Talk:Electoral system/Archive 5

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About Majority Judgment Done

Hi Homunq The Majority Judgment fails the Majority criterion (and hence the Concorcet criterion). This was well known even before Badinski and Laraki put a patent on this method. A discussion of that point, and examples were provided several years ago on the sites of Range Voting. I was typing a counter-example when you reverted my edit. The example is very simple: Two candidates, A and B, and 99 voters. 49 voters give grade 6 to A and grade 5 to B; 1 voter gives grade 3 to A and 4 to B; 49 voters give grade 2 to A and grade 1 to B. The median grades are 2 for A and 3 for B so that B is elected according to the so-called "Majority Judgement". But 98 voters prefer A to B. Johncyclopedist (talk) 15:32, 9 April 2012 (UTC)

@Johncyclopedist: Please use a new heading for every new topic This time I did that for you. --Arno Nymus (talk) 17:53, 9 April 2012 (UTC)

Thanks, Arno, Nymus. I am not familiar with this technology. Sorry also for the typo in my counter-example: the median grades are 3 and 4 and not 2 and 3 as I wrote, but this does not change the conclusion. The point is important: The Majority Judgement fails all the majority criteria which are used in the table for the sake of comparison with other methods, and any sensible definition of "majority". Johncyclopedist (talk) 08:54, 10 April 2012 (UTC)

You're welcome. You can create a heading by adding two equal signs at the beginning and the end of the line: "== About Majority Judgment ==" (the link in my last comment explains that). And please don't add a separator line within a topic. ;) Regarding the content I defer to Homunq, since he is the one you asked for. --Arno Nymus (talk) 18:41, 10 April 2012 (UTC)

Arno: no need to defer, this is about facts not individuals. But now that I'm here, I'll take it.

Johncyclopedist: The definition of the majority criterion used in this article is that a candidate must win if a majority of voters vote them, and no other candidates, at the top rating possible. In your example, though a majority of voters prefer A to B, they do not rate A at the top rating, so the majority criterion does not apply. (As an irrelevant aside: Majority Judgment uses words, not numbers, for its rating categories.) Your example does, of course, apply to the Condorcet criterion, which MJ is correctly marked as failing in this table. Homunq (talk) 20:41, 10 April 2012 (UTC).

The defition you propose is unsual and, in any case, is not the one of the article. The definition of the article is: Will a candidate always win who... Majority criterion—... is ranked as the unique favorite by a majority of voters? Mutual majority criterion (MMC)—... is among a group of candidates ranked above all others by a majority of voters?

These definitions are fine. In my example, there are only two candidates and a majority of voters ranked A as their favorite among A and B. Likewise, A is the unique candidate to be "among a group of candidates ranked above all others by a majority of voters." It follows that MJ fails the majority criterion. The other criterion you mention ("a majority of voters vote them, and no other candidates, at the top rating possible") is logically different. It is an original definition which should not ne labelled simply "majority criterion" but something like "Absolutly best grade majoritarianism". It looks like having been tailored for methods who maximise median grades (and it is satisfied by Approval Voting). In practice your definition will rarely apply since even with two alternatives, I guess it is quite unlikely that a majority of voters give the absolute best grade to one option. The more I think about it the more I think it would be more informative to just say that, contrary to what its name suggest, MJ, just like Range Voting or Approval Voting, is not a majoritarian method. Johncyclopedist (talk) 15:59, 11 April 2012 (UTC)

You say my definition is artificially tinkered to support MJ, and that your definition is natural. You also say that my definition is not the same as the "unique favorite" definition given on the page. Honestly, I'd say exactly the opposite in both cases.

Why do I think my definition is more natural? Your definition is impossible to pass for any method which separately aggregates ratings for each candidate (OK, any resolvable method, to eliminate silly ones like "highest worst rating"). That is to say, any method which passes your definition is going to be principally a ranking, not rating, method, though it can use ratings in some cases (for instance, to resolve Condorcet cycles).

On a practical level: voting is about choosing. In that context, you're trying to draw a distinction between "unique favorite" and "top rating". To me that seems silly on the face of it. For instance, I absolutely do not agree that "even with two alternatives, ... it is quite unlikely that a majority of voters give the absolute best grade to one option."

On a meta level: obviously, the two of us see the world in different ways. Though we agree on the underlying math, we disagree on how the "majority criterion" should be defined. Unless one of us can argue the other one around, which we have to assume is unlikely, the way we solve this on Wikipedia is with citations. I am going to be a stickler, here, and insist that any citations you use come from a source which explicitly considers at least one cardinal method. As you know, the two definitions are synonymous for ordinal methods, and so it would be unfair to read too much into the precise wording used in a source which deals only with ordinal methods. After all, even on the wording "unique favorite" used in this article, we disagree on the application to cardinal methods. (And each of us, until confronted, saw our own interpretation as the obviously-correct one, which any reasonable person would agree with... but here we are, two reasonable people, disagreeing.)

I'll start out the citations with the following: (just a moment while I get the source... hmmm, the source I was thinking of doesn't say what I thought it did, or indeed anything relevant to this question, and I'm still looking for something using scholar.google.com, but it's harder than I thought.) Homunq (talk) 16:45, 13 April 2012 (UTC)

And by the way: I commend you for your forbearance in not edit-warring even though I reverted you. While the current status of the page is the one I support, I would not object to you changing the relevant cells to white/"Disputed" while we resolve this dispute. I would object to you setting them to red/"No". Homunq (talk) 17:15, 13 April 2012 (UTC)

OK, found something. The distortion of cardinal preferences in voting, A Procaccia… - Cooperative Information Agents X, 2006 says: "Majority criterion: [∃j ∈ C st |{i ∈ N : li j = 1}| > n/2] ⇒ F(≻) = j". Note the "j=1" there: that means top possible rating, which agrees with the definition I'm arguing for here.

That's from result #31 of a google scholar search on '"majority criterion" voting', and, while I didn't actually read the first 30 results cover-to-cover, I did pull the most-promising 9 of them, and this is the first clear mathematical definition of any kind that I could find. Really, honestly, I'm not pulling a fast one here by leaving out some earlier result that goes for your side; even the earlier result by James Green-Armytage, a guy who's clearheadedness I would have expected to lead to a workable definition, only defines the MMC and not the MC. There were one or two papers (didn't read the second one enough to be sure) which spoke of "majority criterion" but defined the thing that this article calls "majority condorcet criterion", but that is not germane to our debate here, since we already have that criterion in a separate column. Homunq (talk) 17:30, 13 April 2012 (UTC)

I see you've blanked out the cells with "Disput.". That's fine as a temporary solution, as long as the discussion here is progressing. I presume you're working on your response here. Homunq (talk) 13:20, 14 April 2012 (UTC)

I have looked at the paper you mention, by Proccaccia and Rosenshein. This is an adequate reference since this paper is precisely about the relation between (in their words) "ordinal and cardinal preferences". The quotation you give:

"Majority criterion: [∃j ∈ C st |{i ∈ N : li j = 1}| > n/2] ⇒ F(≻) = j".

is exact, but your comment is not. You write:

"Note the "j=1" there: that means top possible rating"

But this is false. This piece of notation, in this paper, is defined at the second line of the page 319:

"We denote by lij the position in which candidate j is ranked by voter i"

Therefore lij=1 means that the rank is one, which means that candidate j is in the best possible position *relative to the other candidates*. Consequently, the reference you quoted is precisely backing my point. The English writing of the above piece of algebra is: Candidate j wins as soon as the number of individuals who put j in the first position in their rankings of candidates is larger than half the population size.

I understand that the vocabulary "Majority Judgement" is a bit confusing, but things are what they are, and I hope you are now convinced. I suggest to now change this famous cell from "Disputed" to "No". Johncyclopedist (talk) 16:40, 14 April 2012 (UTC)

I want to reply in two points:

I don't think that the name "Majority Judgment" is too confusing. It has been chosen to clarify that it assigns each candidate with the grade a majority approves for that candidate. So it assigns the "Majority Judgment" to every candidate. Obviously, this does not mean that it matches every other usage of the word majority.
Also, I don't think that the above is a good source since it is not a paper for ordinal and cardinal voting systems. It is instead a paper about measuring the distortion resulting from using solely ordinal voting systems while assuming that "voters" (in fact agents in the paper) have cardinal (utility) opinions about the candidates. Since they defined voting systems to be (mandatorily) ordinal, they defined the criterions to be applicable for ordinal voting methods (and not cardinal ones). Thus, it does not match the premise not to be "a source which deals only with ordinal methods"

So, I would be very interested in a paper that actively uses cardinal methods and has a definition of the Majority criterion that makes sense for cardinal voting methods. In fact, it is a little bit embarrassing if a method "fails" a criterion, because the method uses additional information that the criterion just ignores and thus the method "fails" the criterion, because it chooses a "superior" winner. So, for applying MC on MJ, we should look for a definition that is defined also having cardinal voting methods in focus. --Arno Nymus (talk) 18:00, 14 April 2012 (UTC)

Johncyclopedist: You are right that the paper doesn't say what I thought it did. After sifting through over three pages of google results, I was too hair-triggered in posting the first definition I found that appeared to me to be relevant.

However, looking at the reference again, I find I now agree with Arno Nymus.

I understand that if this were debating club, I would have just lost badly, with my own reference used against me. Touchée. But I don't think that Wikipedia is about scoring points (nor am I implying that you think so).

Give me some time to look for another relevant reference. Let me know if you find anything yourself (either of you) Homunq (talk) 18:41, 14 April 2012 (UTC)

I give up for now. I sent email to a couple of people to see if they had a useful reference, but I'm not looking anymore. Way harder than I expected. Homunq (talk) 22:12, 14 April 2012 (UTC)

By the way, I changed the value for MJ/MMC to a light-pink "No" with footnote. I never disputed that value, I don't know how it ever got to "Yes". Must have been an oversight. Homunq (talk) 01:23, 16 April 2012 (UTC) ps. I apologize for reverting you earlier on MJ/MMC, though we still disagree on MJ/MC. Sorry.

Hello. I feel we are coming closer to the conclusion. I looked at quite many references and everywhere I found that the word majority winner, or majority criterion, or majority preferred choice, is used in its usual, relative sense. I mean that in English, the definition: "If a majority (more than 50%) of voters consider candidate A to be the best choice, then A should win" should be understood as "the best among the alternative considered choices." It cannot be understood as "receiving the absolute best evaluation among all the grades of the language". Therefore I maintain that we should change the cell to a full "No". Johncyclopedist (talk) 21:18, 16 April 2012 (UTC)

That's not how I'd understand it. I understand "best" as "best grade". Until we have a source that specifically relates the majority criterion to a rated or graded system, I think we're going to have to leave the cell blank. Homunq (talk) 00:22, 17 April 2012 (UTC)

OK I found such a reference. In the book "Welfare Economics and Social Choice Theory", by Feldman and Serrano (2nd edition, Springer, 2006) page 254. The framework is one in which alternatives are numerically rated and, as the authors put, "it makes good sense to say something lile `person i prefers x to y' or 'u_i(x) > u_i(y)." Then the section "The Majority Voting Criterion" goes on defining the majority voting criterion in the simple choice between two alternatives, and then study the problems arising with more than two alternatives (by the way, those are the interesting problems!). With no surprise, the majority criterion applied to two alternatives x and y deals with how individuals compare x and y, not with the two questions "how many people give to x or y the best possible grade." Johncyclopedist (talk) 14:15, 17 April 2012 (UTC)

OK, I can't check that reference online (or physically, given that I live in Guatemala, and am unlikely to find that book in a nearby library). Still, assuming good faith, I have to accept it as valid, even though it conflicts with my naive assumption of what the MC "should" mean. So I guess we have to change the page. I'd still argue that this cell should be light pink, with a footnote that gives both your reference and the alternate interpretation. I'll let you do the honors of fixing the article; or if you don't do it in the next week or so, I'll probably do it myself. Homunq (talk) 16:26, 17 April 2012 (UTC)

By the way, the definition you use (which I claim to be unusual) which is now: "If a majority (more than 50%) of voters give to candidate A the best rating, then A should win" is not statisfied by the best median, under any sensible tie-breaking rule, thanks to this example: Three possible grades: 3,2,1. Two alternative A and B. Three voters. The grades given to A and B by the three vters are (3,3), ((3,2) and (3,1). Here two thirds of the voters give B the best possible grade, but (after tie breaking) A is chosen versus B. I do not know if this example will convince you. Personally I think it is anectoctical whereas I think that my previous example is important, and deserves that Wikipedia carries the clear message that the method of comparing median gradess, so-called Majority Judgement, does not pass the majority criterion. Johncyclopedist (talk) 14:44, 17 April 2012 (UTC)

I think there's an error in your example; as far as I can see, A gets a unanimous median grade of 3, while B does not have a majority in any sense and has a median of 2. Perhaps you meant something like: 2x(3, 3), 2x(3,2), 2x(1,3), 1x(2,1), where A has a 4/7 majority of top ratings, and a different 3/5(+2 indifferent) plurality preferred over B, but B is chosen? In that case, I'd dispute such example because the majority top-rating A and the plurality preferring them over B are different. In other words, the example hinges on the (3,3) votes which mutually top-rate both; in fact, both have a majority of top-ratings. Homunq (talk) 16:26, 17 April 2012 (UTC)

Regarding the source: I don't see that the definition of "Majority voting criterion" of Feldman resembles the "Majority criterion". Instead, it describes "Majority Rule" and then explains that the relation obtained from that is not transitive and therefore, the Condorcet paradox can occur. It does not mention something like a majority that have a unique favorite.

The perfect think would be a paper on cardinal voting methods that defines the Majority criterion. But since the ranked Majority criterion is considered rather uninteresting for cardinal voting advocates, it is to be expected that they don't use much time on that criterion.

Nevertheless, thanks for searching for it, please don't give up the search.

Regarding the example: the Majority criterion states "if one candidate is preferred by a majority of voters, then that candidate must win"

I think there is a problem in the definition, that - commonly - we use "prefer" to describe a relationship between exactly two candidates. Here, it is used to set one candidate apart of all other candidates. However, I think the three interpretations discussed here are: "A is preferred" means...

I) ..."A is solely topranked" - MJ satisfies the criterion

II) ..."A is solely ranked first" - MJ fails the criterion

III) ..."A is (not necessarily solely) bestranked" - MJ fails the criterion

For ranked systems, I and II are identical. However, III is not only failed by MJ (as your example show), but also by every other method that allows equal ranks. Just assume, all voters vote A=B > C. A and B would have 100% majority, so both must win uniquely (this example can be made more elaborate, but I think, everyone got the point). So, III is surely not the interpretation we are looking for. --Arno Nymus (talk) 18:16, 17 April 2012 (UTC)

Thank you, Arno Nymus. Do you have access to the source? Have you found it online?

In light of Arno Nymus's arguments, I am withdrawing my endorsement for a light-pink "no". This is consistent with my earlier statements: note that above I say that one of the sources I read actually defines Majority Criterion as something I'd call the Majority Condorcet Criterion, and that I consider that irrelevant to the current discussion. Given what Arno Nymus says here, I think that the same thing happened in the Feldman source.

So, my position is now back to what it was before the Feldman source: that we're going to have to leave this with a white "Disput." until we find a source that explicitly relates the criterion to at least one non-ranked system.

Also, I agree with Arno's analysis of Johncyclopedist's counterexample (ie, the part about I, II, and III).

Here's hoping we get a source soon... and a sincere apology to Johncyclopedist. Good work finding the source, which can't have been easy; then I say "you win"; then I take it back. I can imagine that that must be frustrating, so I'm sorry. Homunq (talk) 18:53, 17 April 2012 (UTC)

Sorry for the typo in my example. I mean:

The grades given to A and B by the three voters are (3,3), ((3,2) and (1,3).

(I previously wrote (3,1) instead of (1,3).

I basically agree with the distinction I-II-III made by Arno, but now see my problem: There we have an interpretation (I) which I claim to be unsusal and for which I consider the words "majority criterion" to be misleading. You cannot find a single reference to back your point. We even have difficulty for expressing the idea without ambiguity: for instance Arno uses the word "topranked", but this word is not very good in that case since the word "rank" usually conveys the notion of relative evaluation, whereas what we mean is that an alternative obtains a label (the best one) in itself, not by comparison. I guess that we should use another word for this property (if we are interested in it, which I think we should not be). This property could be called something like the "majority perfectness criterion". It says the following:

An alternative is "perfect" for an individual if it is given the best possible grade. An alternative is "majority-perfect" in a population if it is perfect for more than half of the population members. A voting systems using grades satisfies the majority perfecteness criterion if it choses a majority-perfect alternative whenever the set of available alternative contains one.

Then the method of the best median grade satisfies the criterion. The criterion even looks rather ad hoc for this method. But this criterion does not applies to the other voting rules because with relative data we cannot say if an alternative is perfect or not. If we were to put it in the Table, most cells of the collumn would be impossible to fill. Indeed this criterion is not a criterion for comparing voting rules, as the Table is supposed to do. Thus there is no place for this criterion in our discussion.

On the other hand there is out there a criterion which deserves to be called the majority criterion and which is phrased in terms of ordinal preferences: A rule satisfies the Majority Criterion if it picks an alternative whenever, for more than half of the population, the alternative is prefered to any other available alternative. The important point is that this criterion, which is phrased in ordinal terms, is meaningful for grading systems too, since any grading induces a well defined preference binary relation. With this definition, we can compare all the rules of the Table in a meaningful way, the only exception being Approval since, by design, the approval "language" is so poor that the above construction is not very interesting.

Notice that finding the Feldman-Serrano reference was very easy, because in Economic Theory the story often starts with numerical grades, before talking of Majority Voting. I can find other similar references. In all cases, for two alternatives it will be clear that the Majority criterion is not what I called above the "majority-perfecteness" crietrion and that, accoring to the usual meaning of the words, we have to admit that "MJ fails the majority criterion". Johncyclopedist (talk) 21:16, 17 April 2012 (UTC)

But the Feldman-Serrano reference is not Arno's II, either, since it explicitly applies only to the two-candidate case, and there is in fact a discussion of how it would be problematic to extend it to more candidates. So the only reference we have is the one I gave (Proccaccia and Rosenshein), which, I admit, does define II.

And the Proccaccia and Rosenshein article does not consider rated systems, yet it is specifically about the problems of throwing away rated information in applying ranked systems. It would be sadly ironic if the definition used in that article were used to argue against rated systems which do not throw away that information.

Because that's why I continue to feel that definition I is natural, not ad-hoc. A pure rated/ranked system that passes definition II is impossible. Furthermore, definition II, unlike definition I, is incompatible with other important criteria like IIA and FBC. Yes, some of the criteria in this table are inevitably incompatible with each other; but there's nothing ad-hoc about wanting to keep that incompatibility to a minimum.

So I expect I'll be sticking to my guns until I see a reference that's essentially synonymous with either I or II, and is directly applicable to a rated or graded system (because in the context of ranked systems, as we agree, I and II are synonymous with each other). Sorry. Homunq (talk) 21:52, 17 April 2012 (UTC)

Smith, 2006: "Descriptions of single-winner voting systems" describes both, ranked and rated voting methods. It defines the Majority criterion as follows:

A candidate uniquely top-ranked by a majority of ballots, is elected. (page 27)

On page 30 there is a compliance table, in which "Median rating" is marked as satisfying the criterion. --Arno Nymus (talk) 22:21, 17 April 2012 (UTC)

Well, I'd say that settles it, then. There's no question that this reference is more germane to the issue than Feldman-Serrano or Proccaccia-Rosenshein. Johncyclopedia, any objections to setting this cell back to "Yes"? Homunq (talk) 22:27, 17 April 2012 (UTC) ps. I mean a light green "yes" with footnote, of course.

Hi. I do object to setting the cell back to Yes. About the reference by Smith: as you say this reference is about "ranked and rated" systems and defines a "majority criteria" with respect to "top-ranked", not "top-rated" candidates. What you were looking for is a reference in which "majority" is meant as being "top-rated" by half of the population, not "top-ranked". The Smith reference does the opposite. (This reference seems to be some kind of draft that is floating on the net. It gives no proof of the announced result that "Median rating" is satisfying the criterion, and I gave a counter-example, with again apologies for the mistake). So we are back to my point: I claim that defining "the majority criterion" as a requirement about an alternative getting often an absolute best evaluation ("top rating") rather than a requirement about an alternative getting often an relatively best evaluation ("top ranking") is unusual and confusing. I also would like you to react to my yesterday comments. As to further references about Majority criteria and maximising median grades, I suggest the following: The Majority Judgment Voting Procedure:A Critical Evaluation, by Felsenthal and Machover (easily available on the net). Cheers, Johncyclopedist (talk) 20:58, 18 April 2012 (UTC)

OK, several points:

"Top-ranked" or "top-rated": Though I don't actually think so, my visceral reaction is that you're being deliberately obtuse here. It's perfectly obvious to me that Smith is using the former term to refer to rankings or ratings equally. He had to choose one of the two, and either way someone could have played gotcha. It's the same situation as when Arno said "topranked" above.
Is the Smith ref WP:RS? Good question. I note on Google Scholar that it is "cited by 1", so it's not entirely outside the realm of academic discourse; but you're right, it probably shouldn't be the basis for taking a position in the article. So I'm back to thinking "disput." is best for now.
You say that definition I is unusual and confusing. I (and I believe Arno Nymus) disagree, and say that on the other hand definition II is unnecessarily incompatible with other important criteria. I don't think either one of us is going to budge here, nor do I think it's likely that a third opinion is likely to propose anything but a compromise. And "disput." seems to be the cleanest, most obvious compromise.
React to yesterday comment: see point 3, basically same reaction.
Felsenthal and Machover: they refer to "majoritarianism", without defining it; but it appears to agree with your preferred definition II. But they also do admit that "Majority judgment is majoritarian in the rather narrow sense...". I'd say that even if we were to take this one source as gospel, and ignore the fact that they don't say "majority criterion", it would still argue for a pink or white, not red, cell.
(I also have my own critique of Felsenthal and Machover, but I realize it's WP:OR)

Homunq (talk) 21:19, 18 April 2012 (UTC)

Apparently, Homunq answered at the very same time while I was writing this. Please excuse, if some points concern the same claims.

What I was looking for was a paper for ranked and rated systems, that defines the Majority criterion. And surprisingly, I found one: Smith. It resembles interpretation (I), although he uses "uniquely" instead of "solely", which I used. I understand, that "uniquely top-ranked" is probably not the most elegant formulation possible for definition. However, taking in consideration that our only source on the main site for the Majority criterion is a very biased IRV-fansite, I am more than pleased with the seriousness of the Smith paper. It is also completely understandable, that he used the intuitive "uniquely topranked" instead of "uniquely toprated", since the paper is not only for rated systems and easier to understand in that way (also in the paper he generally didn't use the term "ratings" but entries of a real N-vector, so obviously he couldn't use "toprated").
Please note, that Smith's definition uses the term "uniquely top-ranked". In your example, no candidate is uniquely top-ranked by a majority. Thus, it is not a counter-example using Smith's definition (interpretation I). Please also note, that with interpretation II ("solely first-ranked") still your example does not contain a majority for any candidate. It is only a counter-example for interpretation III, of which - I think - we agree that it is not a correct description of the Majority criterion. Please ask for further explanation on that, if you like.
You wanted me to react to your "yesterday comments". I'm not sure, on which part of that you want a reaction, so I try some of them and please feel free to guide me to the correct one, if I missed it:
The Feldman paper: That text was not about the Majority criterion. Please clarify, whether you agree that, because of this, it is not a source at all (for our problem).
The Felsenthal paper: I didn't see that they mentioned the Majority criterion. Do you have a pagenumber, on which I have to search for the criterion?
About the expressiveness of the criterion: It is common, that a criterion, that is defined for a very restrictive set of systems, can be interpreted in several ways for a superset of the original set - each with the same meaning for systems within the restrictive set, but with different meanings for the bigger set. Mostly, the interpretations range from nearly unsatisfiable for systems outside the original set to easily satisfiable for the systems outside the set. Now, a meaningful interpretation in between have to be chosen. I think, choosing a meaningful interpretation means to maintain the spirit also for the systems not in the original set.
In the case of the Majority criterion, interpretations (I) and (II) are two possible interpretations and I want to say clearly, that both are covered by the formulation of the ordinal case. Note, that - apart form wording - I think, that we agree about what interpretation (I) means, whether we call it topranked or toprated or the highest possible grade or whatever.
About the expressiveness of the second interpretation (II) my opinion differs from yours. Every rated system that uses the additional information of its ballots must fail interpretation (II). So, for a rating system, to fail the Majority criterion (II) would be a good thing. I don't find it in any way a meaningful or reasonable thing, to restrict a criterion in a way to just "punish" a voting system for using all the information it gets. so, (II) would be the "nearly unsatisfiable for systems outside the original set" interpretation. So, it is definitely not a meaningful choice.
Interpretation (I) is satisfied by Majority Judgment, but not by Range voting. So, it allows a differentiation between at least some methods outside the original set and is therefore, at least, not the "easily satisfiable"-case. Also, in my understanding, the Smith paper defines the Majority criterion in exactly this way. Thus, it not just seems reasonable, but also is the definition in the only source we have about the Majority criterion in respect to application to rated systems. --Arno Nymus (talk) 22:42, 18 April 2012 (UTC)

I agree with everything Arno Nymus says above except paragraph 8 ("About the expressiveness..."). There are some systems, such as James Green-Armytage's cardinal-weighted pairwise, which are essentially ranked systems, but use ratings information in certain limited ways; and these can (and usually do) pass the majority criterion according to interpretation (II). So my problem with interpretation (II) is not that it is impossible for a rated system to pass it, but that it is impossible to pass (II) and the Independence of Irrelevant Alternatives (IIA) criterion or the Favorite Betrayal criterion (FBC). (The proofs of these incompatibilities are essentially similar to the well-known proofs that IIA and FBC are incompatible with the Condorcet criterion). Since the majority criterion, by interpretation (I), is indeed compatible with IIA and FBC (as demonstrated by the fact that MJ passes all three), I find interpretation (I) to be more natural and useful. Homunq (talk) 02:45, 19 April 2012 (UTC)

Hello. Re-reading our comments, I feel that we do not have a great quality discussion, despite the fact that, as far as I can see, we agree on most things. We have come to a point where we discuss whereas the intended signification of one word in a old working paper (Smith's piece) matches our understandings of a phrase ("The Majority Criterion") like if that was God spell. We should instead try to find the best ways to synthetise known facts and concepts (about which, I repeat, we seem to agree). I am very busy those days but I would like to take a few days to think through the matter. Johncyclopedist (talk) 20:07, 20 April 2012 (UTC)

Here's an idea: in the definition of the MC, define both ranked and rated versions, and then treat this column like the IIA/ISDA column; that is, have approval and MJ say "rated" in light green. In fact, this idea seems like such a good compromise to me, that I'm going to be bold and implement it, to see how it looks. Homunq (talk) 17:34, 21 April 2012 (UTC)

I'm quite happy with this solution overall, although the wording perhaps needs some work. Johncyclopedist, if you like it as well, I owe you some thanks for spurring me to find a better solution, rather than just continuing to argue back and forth here on talk. Homunq (talk) 18:04, 21 April 2012 (UTC)

Done Homunq (talk) 13:52, 28 April 2012 (UTC)

RfC: Is later-no-harm applicable to Approval voting? Done

[request for comment removed]Filingpro (talk) 01:37, 20 April 2012 (UTC) SUMMARY: Is the later-no-harm criterion applicable to Approval voting, or can't we say either way due to lack of WP:RS specifically addressing this question? Proponents of non-applicability claim that preferences preclude approvals by definition, while proponent of application claims this interpretation leads to a mathematical contradiction while application is trivial. Two prior discussions precede above re: “Approval/LNH “. (Also, separate active debate relating to same article: see Talk:Voting_system#About_Majority_Judgment) Filingpro (talk) 23:26, 18 April 2012 (UTC) with clarifying edits to summary by Homunq (talk) 03:29, 19 April 2012 (UTC)

Note: I made this new section because I understand that the last one was too big but I think a whole new page is a really bad idea. Homunq (talk) 20:54, 17 April 2012 (UTC)

Thank you to Contributors – I have responded to Arno Nymus new comments above, and more importantly suggested a new proposal below. I would also like to propose starting a new ~~page~~ section, "Proposals For Approval/LNH" which references the prior two debates and simply summarizes the current proposals, so then anyone could choose comment.Filingpro (talk) 20:36, 17 April 2012 (UTC)

NEW PROPOSAL

To contributors, I propose the following change to the article based on our collective efforts:

BEGIN EDITS…

Compliance Chart Approval/later-no-harm: NO [with footnote to below...]

[footnote] A voter changing an approval for candidate 'A' to the approval of both candidates 'A' and 'B' harms the probability of 'A' winning. (see Woodall’s definition of a local and relative criteria [cititation to Woodalls document]).

END EDITS

-Homunq this satisfies your requirement of abundant simplicity.

-It satisfies Wikipedia’s requirement of citation, which I believe is all that is relevant.

-It satisfies that we are making no original material by any means, only taking the direct, clear meaning of Woodalls definition of harm.

The argument that “later” does not apply to approval has no relevance because in approval the order of the candidates is irrelevant, so any of the candidates voted on can be the later candidate.

Please offer suggestions, improvements. Thank youFilingpro (talk) 20:36, 17 April 2012 (UTC)

Ummm.... no, though I suppose I might eventually agree with something similar to this.

Basically, there's a controversy over how the math applies. I do NOT accept that your proof is valid or that it is the right way to apply Woodall to Approval. The statement you suggest for the footnote is true, and I can see the argument that it is relevant, but in order for me to be OK with this it would have to at least explain why this is controversial. And if there were an explanation of the controversy, then, for fairness's sake, we'd have to change the cell color from red to pink. And then there's the whole question of finding citations for the controversy, because I really do hope to get this article back up to FA status some day.

So I'd council you to quit while you're ahead; you've gotten the page to change from red "NA" -> red blank. If you keep pushing, and if you find further relevant references (and I mean facially relevant in themselves, without requiring pages of "proofs" on a talk page), you might get it changed to a pink "no" with footnote. I think that would decrease the quality of the table overall (which should ideally have as few pink and light green cells as possible, except in the cases of the ballot type and ISDA columns where the other colors have a single clear meaning).

ps. Thanks for adding this to the bottom of the page, and please continue to do so; scattering replies all over the section makes it really hard to read. Homunq (talk) 20:50, 17 April 2012 (UTC)

NEW RELIABLE SOURCE From Woodall, Voting Matters - Issue 6, May 1996, http://www.mcdougall.org.uk/VM/ISSUE6/P4.HTM ...

Scoring methods that use ~~addition~~ [simultaneous addition of voters' scores] to determine the winner violate later-no-harm:

Woodall states, "Point Scoring (PS) methods are those where each candidate is given a certain number of points for every voter who puts them first, a certain (smaller) number for every voter who puts them second, and so on, and the candidate with the largest total number of points is elected. These methods have very similar properties to FPP, although later preferences can now count against earlier preferences, so that later-no-harm fails, and mono-raise-random and mono-sub-top also fail in most cases."

~~The axiom of addition now determines the change in probability of outcomes (i.e. harm), thus approval is unmistakably implicated.~~ Filingpro (talk) 10:09, 18 April 2012 (UTC) The simultaneous addition of a voters’ scores causes the change in probability of outcomes (i.e. harm), thus later-no-harm must similarly apply to approval.

No. Even if one were to accept your conversion of truncated preferences to approval, which I don't, approval still doesn't give a "(smaller)" number of points for second than first place. The passage in question obviously does not apply. Homunq (talk) 11:10, 18 April 2012 (UTC)

Yes the passage in question does apply because it proves later-no-harm applies to any scoring system and it is a trivial matter that approval is a scoring system because it assigns assigns a “0” or “1” to a candidate depending on the optional approval of the voter. The plain language of applicability as “capabable of being applied” affords you no such license to exclude approval from application, when it is prima facie that approval is a mathematical subset of a scoring system by ~~the axiom of addition~~ its relying on simultaneous addition of voters' scores, while those scores are merely limited to 0 or 1. Later-no-harm cannot therefore apply to scoring systems and also not apply to approval because this interpretation leads to an obvious mathematical contradiction. The interpretation you adhere to therefore has no mathematical basis, while it maximally pollutes the criterion’s ability to distinguish voting methods and damages social choice theory. I am now obligated to seek other means of resolution to this matter. Filingpro (talk) 16:38, 18 April 2012 (UTC)

@Arno Nymus: Would you kindly express whether you concur or reject my position so I may make the appropriate determination as to whether Wikipedia Third Opinion applies? Thank you Filingpro (talk) 16:51, 18 April 2012 (UTC)

It is my expressed opinion that the difference between a red blank, pending a clearer source citation, and a red no, would not worth the time you're devoting to it, even if your position were correct. That is to say that even granting every one of your arguments, there would be no compelling reason that a blank would be unacceptable; but if I am right and the "No" is WP:OR (and, incidentally, wrong; which is to say its a biased view of a situation that admits multiple interpretations) then a "no" is indeed unacceptable. Therefore I think a third opinion would be a waste of time for all involved. However, I understand that you do not feel this way, and if you can't change your mind on this, then it is indeed appropriate to ask for a third opinion. If that is the case, please let's do it now and get it over with. Homunq (talk) 17:07, 18 April 2012 (UTC)

@Filingpro: I kindly express that I reject your position on Woodalls cite on "Point scoring methods". Please note, that Woodalls definition shows, that his "Point scoring methods" are what we call a strict positional voting system (e.g. Borda count). [Especially note, that Woodalls "Point scoring methods" is NOT what we today understand as "scoring method" (e.g. Range voting).]

Approval is NOT a strict positional voting system (or Point scoring method as Woodall calls it in the cite), since Approval does not give each candidate "a certain number for every voter who puts them second" and so on. In fact, being the second best candidate can result in different values for different voters in Approval, depending on whether the voter approves that candidate or not. --Arno Nymus (talk) 18:03, 18 April 2012 (UTC)

Approval need not be a strict positional voting system when it is prima facie that the ~~axiom of mathematical addition~~ simultaneous addition of scores causes the changes in probability outcomes and therefore the application of later-no-harm. If later-no-harm applies to strict positional systems on this basis, it must also apply to approval if the same axioms of addition apply, to which they must apply, of course.
To show applicability all that is required is to show one case: when I change my vote from approval of candidate A to approval of A and B, then I harm the probability of A winning according to the definition of harm by Woodall. Meanwhile it has been shown ~~by the axiom of mere mathematical addition~~ that later-no-harm unequivocally applies to approval in all cases.
Lastly, proponents on non-applicability have not demonstrated how the plain language in the definition of the word “applicable” –i.e. “capable of being applied” precludes the trivial application of later-no-harm to approval.

I remain obligated to seek other means to resolve this matter.Filingpro (talk) 21:52, 18 April 2012 (UTC)

This full argument is overwhelming to me. What I know is Later-no-harm criterion is a criterion that seems to only apply to ranking systems, not rating/scoring systems. The intention of the criterion is to encourage voters to offer as many preferences as they have, and not fear lower preferences will harm the chances of winning for the higher ones. If voters fear this, many will bullet vote, and we're back to plurality for those who refuse to support more than one. I can see how a person might say approval satisfies it, given it prevents voters from offering rankings, so their compromises are clear - they want anyone in a set of approved candidates to beat anyone in the set of disapproved candidates, but I'm sure that's not the intention of the later-no-harm criterion, and there's definite harm to "approve" because the more candidates you support, the more your help your favorite lose. Tom Ruen (talk) 22:56, 18 April 2012 (UTC)

ad 1) Filingpro wrote "when it is prima facie that the axiom of mathematical addition causes the changes in probability outcomes and therefore the application of later-no-harm."

Reply: Even if someone want to understand this in the most meaningful way, it is obviously wrong and Woodall does not say anything like that. Please don't use the term "prima facie". If something is really "prima facie", it does not have to be said about it.

But also note, that your sentence is not even clearly defined. You would have to say at which particular place the "mathematical addition" is used. Note, that IRV uses addition to sum up votes and it satisfies LNH.

Filingpro wrote it must also apply to approval if the same axioms of addition apply

Reply: Which they don't, if it is understood in the only meaningful way [i.e. in relation to positional voting systems].

ad 2) Filingpro wrote "when I change my vote from approval of candidate A to approval of A and B, then I harm the probability of A winning"

Reply: Right, Approval violates equal-no-harm, which is also violated by e.g. the LNH-compliant Minimax pairwise opposition method.

Filingpro wrote "it has been shown by the axiom of mere mathematical addition".

Reply: So, could you please just mathematically define this ominous "axiom" you are talking about?

Please see response much further below in new entry on far left titled "Re: Arno Nymus objection to word axiom".Filingpro (talk) 09:37, 19 April 2012 (UTC)

ad 3) Woodalls definition [of LNH] demands that on the ballot a later preference have to be specified. A later preference is a candidate, such that another candidate is preferred over that "later" preference. Your "trivial application" proposal changes the underlying relation by encoding the approval ballot "{a, b}" as the strict ranking "a > b" or "b > a". Thus, your encoding adds the relation between a and b, so that after that you can say that b is "later" than a; equally you can say that now a is later than b. So, by your encoding you just added the justification to call the other preference later, although before your encoding this justification wasn't there. Above I showed you that by such kind of encoding you also can encode IRV-ballots in a way, that preferences between candidates are added, so that in IRV "later" preferences would harm, although the same winners are elected in every case. Obviously, this kind of encoding is "perverse", but it shows, that this is not at all an "trivial" applicability. --Arno Nymus (talk) 23:42, 18 April 2012 (UTC)

Temporary truce on this one? I realize with the encoding issue you are claiming its not a preference ballot I'm creating because the input is approval. Meanwhile I claim that by putting the preference encodings in the ballot in a fixed manner, and then simply making the voter unaware that he/she is marking preferences and instead instruct them to mark approvals, I have created a voting method consistent with Woodall but where its only possible to receive the same voter changes (i.e input) as an approval voting system would. I suggest for the time being, we can agree to disagree on this, while moving to a separate issue next point...
My new argument doesn't use encodings. I'm simply saying now that Woodall gives us a citation that Borda violates later-no-harm, we can trivially deduce that approval also fails, because they both obviously use simultaneous addition of scores. I realize, you disagree because Woodall only mentions strict positional scores, and so you claim I can not "apply" approval.Filingpro (talk) 09:37, 19 April 2012 (UTC)

@Tom Ruen: Thanks for joining the discussion. What do you think about Homunq's compromise proposal (blank out the cell, so that it just has the "failure"-red and since the line to the left "NO"-cell is invisible, the only text in the "merged" cell is the "NO" for ">2 ranks allowed"? --Arno Nymus (talk) 23:54, 18 April 2012 (UTC)

It seems like it should just be NA=not applicable for ALL nonranked methods, including approval. Anything else seems dishonest. Tom Ruen (talk) 00:25, 19 April 2012 (UTC)

The fight should be at Later-no-harm criterion, and even there looks troubled, talking about ratings in the opening. The definition I see from Woodall is: "Later-no-harm. Adding a later preference to a ballot should not harm any candidate already listed." What source shows this criterion being applied to nonranked ballots? Tom Ruen (talk) 00:27, 19 April 2012 (UTC)

As I've said before, I have no problem with either "NA" or blank, but think that "No" is inaccurate.

Note to editors visiting this page because of the RfC: the discussion above in the section Talk:Voting_system#About_Majority_Judgment is also ongoing but somewhat stagnant (only 3 editors participating, and all of them taking essentially the same position as at the start of the discussion), and so additional input up there would be welcome too. Homunq (talk) 01:03, 19 April 2012 (UTC)

"NA" is ok for me, too. --Arno Nymus (talk) 01:27, 19 April 2012 (UTC)

Also, I think that the name of this section/RFC is slightly wrong. The question is not so much whether LNH is applicable to approval; the real question is, can we even answer the question of applicability within the rules of wikipedia. As far as I understand, Filingpro's position is that yes, we can, that it's a simple matter of WP:CALC; my position is that we can't, that answering that question would violate WP:SYNTH (as well as WP:UNDUE, because I think that there's a legitimate argument that Filingpro's answer is itself wrong; so if we included Filingpro's answer, we'd need to include the counterargument as well). Homunq (talk) 02:56, 19 April 2012 (UTC)

@Homunq: it appears you may be the creator or involved with the promotion of SODA to which I hope you are able to publish and perhaps see realized.

Re: Arno Nymus objection to word "axiom": Yes my use of the word axiom was not always precise and so I have made strikethroughs above, so the intended point is now more clear. SUMMARY: To me it remains obvious to a reasonable qualified observer in this field that, the reason Woodall applies later-no-harm to borda is because the voters’ scores are added simultaneously. This is why I say it is “prima facie” that approval violates later-no-harm because scores of 0 or 1 for candidates are also added simultaneously. (Other responses to Arno Nymus above, in our usual friendly fashion) Filingpro (talk) 09:37, 19 April 2012 (UTC)

To be clear I mean in borda when I score A=2, B=1 my score is added to B at the same time (simultaneously) as my score is added to A so I harm A's chance of winning, as opposed to IRV with rounds only A is considered first before any consideration for B.Filingpro (talk) 10:18, 19 April 2012 (UTC)

@homunq I very much like the phrasing of your question regarding Wikipedia standards and again I appreciate your are looking for compromise solution that is practical. I concur that this is a very valid question you are raising - I shall explore that and offer my advice while I remain open to yours.Filingpro (talk) 09:37, 19 April 2012 (UTC)

Yes, I am the (principal, not sole) invetor of SODA voting. Thanks, I'd love to see SODA make it into this article and into real-world use. (ps. to all editors: that makes it very easy to figure out my real name, which is fine; but please don't use it here.)

@Filingpro: Thanks for acknowledging my issues with WP:SYNTH. There would be no shame in backing off this question (and, if you want to, canceling the RfC) until we find better sources. Homunq (talk) 11:40, 19 April 2012 (UTC)

@Filingpro: I came to the point that I request you to answer at the end of the post (from now on). Although this means that some advantages of wiki can't be used, it is in fact easier to follow the discussion that way. Thank you.
Woodall's conclusion is for systems that take the mere numbers of voters, that ranked a candidate in i-th position (let's call this #v_i(c)) and multiply these with a certain factor and sum that up, so the score of a candidate will be: $s(c):=\sum _{i\in ranks}p_{i}\cdot \#v_{i}(c)$ . Also, there are additional conditions (corner cases), that Woodall (I think because of simplicity) does not mention at the page you linked. For example, the second position must give more than zero points (this is the reason, why this proof does not apply on plurality), Woodall's conclusion bases on the following generalized counter-example. There are (2k+1) voters, (k+1) with preference list "A > B >...", k with preference list "B". Now, if the (k+1) voters hide their later preferences, A will have $(k+1)\cdot p_{1}$ points, B will have $k\cdot p_{1}$ points. Thus, A will win. If the (k+1) voters express their later preferences, A will still have $(k+1)\cdot p_{1}$ points, but B will now have $k\cdot p_{1}+(k+1)\cdot p_{2}$ points. Thus, if $(k+1)\cdot p_{2}>p_{1}$ , B will win. Since we assumed p_2 to be greater 0, there will always be a k, that satisfies this inequality. Thus, positional voting systems violate later-no-harm.
Now, for Approval to be expressed in the terms of the proof above, equal ranks have to be allowed, thus meaning, $s(c):=\sum _{i\in ranks}p_{i}\cdot \#v_{i}(c)$ can describe Approval, if #v₁(c) is the number of all voters, that approve c and #v₂(c) is the number of all candidates that does not approve c. As can easily be seen, this means, that the certain numbers for each voter that rank a candidate in a certain rank, are p₁=1 and p₂=0. Since the proof above (2.) needs p₂>0, it does not hold for Approval voting (as it does not for Plurality).
Filingpro wrote "the reason Woodall applies later-no-harm to borda is because the voters’ scores are added simultaneously." In "Minimax pairwise opposition" the voters' scores are added up simultaneously from the single voters preferences. Nevertheless, Minimax pairwise opposition satisfies later-no-harm. So, your description is still not accurate. It would have to be more detailed, in fact, what you - as I understand - want to express is, what I describe above in point 2.
However, I am open for new sources. We should search for a source that includes Approval voting. --Arno Nymus (talk) 14:40, 19 April 2012 (UTC)

Excellent suggestion. Thanks for the numbers!
Yes it is self-evident in a scoring method that any positive (non-zero) score given to a second choice harms the probability of any first choice(s). This is also why approval fails later-no-harm.
Approval for a second candidate receives a score of “1”, not “0” so later-no-harm is obviously violated. You will say this therefore cannot be “later” it must be “equal”. I will cite the definition of the word applicable as “capable of being applied” as a "prima facie" case (to which you will object). Therefore, to solve this deadlock I will hopefully start a new topic below.
Minimax pairwise opposition does not “add voters’ scores” because there are no voter scores. Instead it considers the candidate with the least worst defeat in all possible pairwise plurality contests, whereby the results of each pairwise contest is determined by the relative rankings of the candidates, not scores. Filingpro (talk) 21:13, 19 April 2012 (UTC)

FP, I really have no idea what you mean when you keep saying "prima facie". Please, from now on find some other phrasing to convey whatever it is you're trying to say with those words. It would be even better if you could explicitly relate your meaning to Wikipedia policy. Homunq (talk) 21:35, 19 April 2012 (UTC)

ad 4) Obviously, for Minimax it does not apply. I just said that your statement is still too vague to exclude it and therefore it doesn't say anything. I don't agree that it can be seen directly from the word score. But as said, my number [14:40].2. defines it detailed enough.
ad 2) If it would be self-evident, you would have succeeded in formulating it in an explicit way after being ask for it, which you haven't.
However, the Woodall source makes a statement about positional voting systems. If you want to see Approval as an positional voting system, you have to use my number [14:40].3. above, which shows, that Woodalls statement does not say that Approval violates LNH. So, your remarks are not covered by that paper.
ad 3) I don't object, that approving another candidate can harm a candidate. But, this discussion is not about "another-no-harm", which is also failed by Minimax pairwise opposition (if another candidate B is expressed equal than A on the ballot (instead of being not expressed), this can harm A).
Filingpro wrote "You will say this therefore cannot be “later” it must be “equal”." Reply: No, not by that implication which is suggested by "therefore". I object that. Instead, I would say that LNH demands the expression of a later preference on the ballot, whereas this is an expression of an equal preference. Only because you speak of a "second candidate" does not mean that this candidate is a "second" rank candidate. Interpreting Approval voting as a positional voting system requires to have a certain threshold between two ranks. This means, that all approved candidates are first rank candidates and all not approved candidates are second rank candidates.
Filingpro wrote "I will cite the definition of the word applicable as “capable of being applied”" Reply: Why should you do this? "Applicable" is not one of the words you would have to define more clearly - in contrast to what exact additional mechanisms you meant previously (which is now descibed by my number [14:40].2.). The meaning of applicable seems pretty unambiguous in this discussion.
Filingpro wrote "Therefore, to solve this deadlock I will hopefully start a new topic below." Reply: I hope, this means, that you come up with a reasonable source that defines LNH and includes Approval voting. That would be most welcome.
However, what do you think of Tom Ruen's proposal to resolve the issue? --Arno Nymus (talk) 22:28, 19 April 2012 (UTC)

Here is my final suggestion:

Social choice theory is fundamentally concerned with strategy and practical functioning of methods. I would recommend that the editorial gold standard for later-no-harm be “Does this voting system liberate the voter from the strategic question of “bullet” voting?”

Here is what a clear, simple table would look like for common methods…

	Later-No-Harm
Approval	Fail
Borda	Fail
Range	Fail
Condorcet	Fail
IRV	Pass
Plurality	NA

This best serves the public and is a simple standard that best serves the field.

In the meantime, I consent to the proposed compromise “Blank” designation until an acceptable citation can be found for all. Filingpro (talk) 01:37, 20 April 2012 (UTC)

Yay! Consensus, however tentative and provisional! Arno Nymus, slap a "Done" on this section and lets all have ourselves a beverage of our choice, we've deserved it! @Filingpro: *tink* Homunq (talk) 01:59, 20 April 2012 (UTC)

I agree with the "compromise “Blank” designation until an acceptable citation can be found". Thx to all participants. --Arno Nymus (talk) 03:45, 20 April 2012 (UTC)

Done Homunq (talk) 13:52, 28 April 2012 (UTC)

International Institute for Democracy and Electoral Assistance handbook

The International Institute for Democracy and Electoral Assistance made a handbook on voting systems.

Here are copies of the handbook:

And here are summaries:

Fact sheet:

English: http://www.idea.int/publications/esd/upload/Factsheet_orgA.pdf - http://www.webcitation.org/67XAJHNu1

Info pages on various versions:

Info on more diagrams: http://www.idea.int/esd/materials.cfm - http://www.webcitation.org/67XCs3vyM

Electoral systems:

English: http://www.idea.int/esd/upload/ESD%20map-english.pdf - http://www.webcitation.org/67XCxv7lL
Arabic: http://www.idea.int/esd/upload/Map-arabic.pdf - http://www.webcitation.org/67XCzqc6n
Portuguese: http://www.idea.int/esd/upload/ESD%20map-potugese.pdf - http://www.webcitation.org/67XD8YWYa
Spanish: http://www.idea.int/esd/upload/MAPAOK13junio2006.pdf - http://www.webcitation.org/67XDA18cu
Thai: http://www.idea.int/esd/upload/electoral-system-design-map-THAI.pdf - http://www.webcitation.org/67XD3lUav
English poster: http://www.idea.int/esd/upload/ESD%20poster-70x50cm.pdf - http://www.webcitation.org/67XDFggAb
Slides:
- English PDF: http://www.idea.int/esd/upload/slides_vector.pdf - http://www.webcitation.org/67XDIsG0i
- English PPT: http://www.idea.int/esd/upload/ESD%20map%20on%20slides.ppt - http://www.webcitation.org/67XDQUdMU

Handbook PPT:

English: http://www.idea.int/esd/upload/ESD_Slides.ppt - http://www.webcitation.org/67XDVrenJ

WhisperToMe (talk) 15:08, 9 May 2012 (UTC)

IIA and Majority Judgement, Approval, Range Voting

According to the table on this page, Majority Judgement, Approval Voting, and Range Voting all satisfy the Independence of Irrelevant Alternatives criterion. However, I'm not sure this is actually the case. Here's what I'm thinking. I want some feedback before I go and edit this page.

In theory, these should all satisfy the criterion. You take the candidate with the best average ranking, and the average ranking is not directly dependent on any other candidate. So if candidate x has an average ranking of 4 out of 5, candidate y has an average of 3 out of 5, and z has an average of 2 out of 5, you can remove candidate z, and candidate x will still beat candidate y 4-3. Okay, I get that.

But in practice, won't people rate preferred candidates as high as possible and least-preferred candidates as low as possible in order to maximize the power of their votes? For example, suppose I were to vote between Andrew Johnson and George Washington on a 0-99 scale. I would rate Johnson 0 and Washington 99. But if I were to vote between Andrew Johnson, George Washington, and Joseph Stalin, I would give Stalin 0, Johnson 92, and Washington 99. And if it were just between Stalin and Johnson, I would give Johnson a perfect 99. My preferences would not have changed, and I am not even voting insincerely. It's just a matter of what the scale is. The scale is always based on what candidates are in the race.

I ran a very simple study of how this idea might affect an election. Suppose there are three candidates (Ca, Cb, and Cc) and three voters (V1, V2, and V3) who must rate the candidates Excellent, Good, Fair, or Poor in a Majority Judgement election. Voter 1 rates Ca as Excellent, Cb as Fair, and Cc as Poor. V2 rates Ca as Poor, Cb as Excellent, and Cc as Good. V3 rates Ca as Fair, Cb as Poor, and Cc as Excellent.

Candidate C will win, because his/her median is Good. But now lets say Candidate A drops out right before the election. Voter 1 will adjust his scale, and rate Cb as Excellent and Cc as Poor. V2 will also rate Cb as Excellent and Cc as Poor. V3 will not change anything, and will rate Cb as Poor, and Cc as Excellent. Now Cb has two "Excellent" ratings and one "Poor" rating, whereas Cc, the original winner, has one "Excellent" rating and two "Poor" ratings. So Cb defeats Cc without any voters changing their preferences, voting insincerely, or even needing to know how the other voters would vote.

Since a losing candidate dropping out can change the outcome of the election, it doesn't seem fair to say this satisfies IIA. Shouldn't there at least be an asterisk to explain that since the ratings are subjective, they are really subject to change based on what candidates are in the race?

So that's what I'm thinking. What do you all think?

Dr. Hipopotamo (talk) 03:39, 16 May 2012 (UTC)

You are correct that in practice, voters will probably change their ratings for a given candidate depending on who else is in the race. However, if that is so, no (deterministic) system could ever possibly pass IIA. The only way IIA makes any sense at all as a concept is if you define it mathematically, and ignore such practical considerations. On this basis, the systems in question do pass. Thus, while it may be worthwhile to mention your issue on the page, I'd do so in the definition of IIA, not in the table. I'll give it a shot... Homunq (talk) 08:54, 16 May 2012 (UTC)

There already was a parenthetical comment in the definition of IIA on this point; but I've expanded it with a footnote which has reference and further clarification. Satisfactory? Homunq (talk) 09:45, 16 May 2012 (UTC)

Okay, I see what you're saying. The footnote does help. Thank you. My only worry is that someone might already know the definition of IIA and skip straight to the table, where they wouldn't see the footnote. Then, based on the table, the cardinal systems look better, and Condorcet systems look worse by comparison. In my opinion, for example, ranked pairs is better than range voting because it satisfies Local Independence and ISDA, neither of which the cardinal systems can satisfy in practice. But looking at the table, the cardinal systems look better, because they actually satisfy IIA completely. Assuming equal standards across the board, the cardinal system works perfectly, but in practice, it's worse than ranked pairs, which the table doesn't reflect unless you look at the footnote. I don't know much about editing Wikipedia; is it possible to link to the same footnote from two different places (i.e. the definition of IIA and the footnote)?

Thank you for the table, by the way. I'm writing a research paper about this stuff and all of these articles on voting systems, especially that table, have been extremely helpful. Dr. Hipopotamo (talk) 14:17, 16 May 2012 (UTC)

It's good that you're up-front about seeing certain systems as better than others. On a purely personal level, I'd probably debate your points. For instance, while I agree that many aren't going to vote the same on candidates 1 and 2 when candidate 3 drops out, I still think it's good to have a system where they can if they want to, and where everyone can vote the same on candidates 8 and 9 when candidate 10 drops out. But the point is that our personal preferences have no place on the page. I think pushing the footnote into the table would be taking a stance we couldn't justify. Homunq (talk) 14:50, 16 May 2012 (UTC)

The criteria are mathematical statements. If it would be allowed to consider the voters changing their ballots apart from the changes demanded by the criterion while checking for compliance with the criterion, most criteria would be failed by every system - as described for IIA by Homunq. They would all become completely meaningless. Thus, if we add the footnote for IIA, we would also have to add footnotes for most of the other criteria and this would result in maximal confusion for the table and no additional information.

However, I don't think that the cardinal systems look "to good" in the table. In fact, the table shows their advantages and disadvantages, as they show the advantages and disadvantages of the other systems. For example, range voting has the problem of failing all of the absolute criteria, IRV fails all of the criteria about changed voters opinions, Minimax fails the criteria of changing nominations and ranked pairs and the Schulze method... well, in relation to the others, these two look pretty well in the table. So, I definitely don't think that cardinal voting looks to good in the table in comparison to e.g. ranked pairs. --Arno Nymus (talk) 23:39, 16 May 2012 (UTC)

I still think that the best system is the one that works best in real life, but like you said, this isn't about personal opinions. In the end, there is no such thing as a completely neutral article, because you still have to decide which information is considered "important." Since you've done so much work on the article, I'll defer to your judgement on this issue. Thanks! Dr. Hipopotamo (talk) 18:14, 17 May 2012 (UTC)

Random Winner, Random ballot, resolvability and tie-breaking

The compliance table says both random voting systems fail the resolvability criterion. Is it correct? They are usually used as tie-breaking systems, and tie-breaking needs to be resolvable to be really useful.

The table also says random ballot accepts only single mark ballots. Some preferential voting systems which output full rankings use random ballot as tie-breaking, using the ranks to generate a TBRC list. So, random ballot allows ranking, and maybe also scores. --Wat 20 22:19, 24 May 2012 (UTC)

Ranked Pairs vs Schulze method

I split the IIA/LIIA/ISDA column into 3 independent columns. Now Schulze method is Pareto dominated by Ranked Pairs in the comparison table and IMHO it is a very meaninful information. --Wat 20 01:20, 10 July 2012 (UTC)

Runtime complexity

I was willing to substitute those yes/no in the Polytime column with runtime complexity. Plurality has O(N), Schulze method has O(N³), Ranked Pairs has O(N⁴). But I don't know the runtime complexity for all methods. --Wat 20 01:20, 10 July 2012 (UTC)

Hmmm.... For me, the point of "polytime" is, "is it even theoretically possible that no modern computer could ever calculate a definitive result". Even order 4, 5, or 6 polynomials just go up to a trillion for 100 candidates, something that a modern computer could power through. I can see that order 1 or 2 could maybe be done by hand... but I don't know, I think hand-counted and (repeatably, checkably) computer-resolved is as good as hand-resolved. So I don't see the need. If you think this is worth it, though, I could probably help figure out the order for most or all of these systems. Homunq (talk) 04:30, 29 July 2012 (UTC)

For very large number of candidates, differences between runtime complexities matters, even if they are all polytime. i.e. trying to rank tens of thousands of products, O(N³) becomes tens of thousands times slower than O(N²). --Wat 20 21:13, 07 Septemper 2012 (UTC)

Updates To Plurality In Compliance Table For LNH

Plurality and Approval are categorically different voting systems. In a strict approval ballot there can be a multiplicity of candidates marked, whereas in a strict Plurality ballot there can be no more than one candidate marked.

Currently for Approval there is debate over how the multiplicity of approvals can not be mapped 1-1 to ordinal preferences and what this means, whereas in Plurality there can be no such debate because there is no multiplicity of candidates.

NA is used because there are no multiplicity of candidates to consider which is categorically different than a voting system which allows a multiplicity of candidates to be indicated which passes the criteria.

Green is used and should be used whenever a voting system does not fail a criteria. Red should only indicate failure. The fact that other postings in this table are not consistent with this editorial sensibility I can not account for here.
Filingpro (talk) 07:39, 11 September 2012 (UTC)

Good work. But I don't agree with your coloring. I've added a footnote and put the coloring how I think it should be. If you (or anyone) disagree with the second sentence of the footnote, feel free to delete that sentence and set those two cells to white. Homunq (talk) 16:04, 11 September 2012 (UTC)

1. Thanks. I appreciate you are offering some alternative editorial standard as a reason for your coloring.

Q: Aren't the two logically the same - i.e. same meaning?

Your reason for red: "being discouraged"

Your reason for green: "not being encouraged"

Q: If the reason is the same, shouldn't they be the same color?

Filingpro (talk) 23:56, 12 September 2012 (UTC)

"Being discouraged" = failing LNHa = Red

"Not being encouraged" = passing LNHe = Green

Homunq (talk) 00:54, 13 September 2012 (UTC)

2. Also I notice when sorting ascending by Later-No-Harm, Majority Judgment appears above Plurality. Do you agree this is incorrect? If so, what can we do about this? Thanks.

Filingpro (talk) 00:07, 13 September 2012 (UTC)

Better? Homunq (talk) 00:54, 13 September 2012 (UTC)

re: 2. Thanks. Yes. Sorting fixed. Filingpro (talk) 01:49, 13 September 2012 (UTC)

re: 1. I'm not sure I understand yet how you are creating a consistent standard. I remain open.

Q1: To start can you answer, are these semantically the same meaning (A) "Being discouraged" (B) "Not being encouraged"?

Q2: How do you know which standard - i.e. A or B to apply?

Q3: How does A apply to LNHe?

Q4: How does B apply to LNHa?

Q5: Are you selectively applying different standards (i.e. A or B) to different criteria? Please explain.

Thanks.Filingpro (talk) 01:49, 13 September 2012 (UTC)

Q6: For "not being allowed to mark later preferences" doesn't Plurality always satisfy both A and B?

Filingpro (talk) 02:09, 13 September 2012 (UTC)

A6: Pretty much. That is, "not allowed" isn't exactly the same as A or B but similar.

A1: Again, pretty much.

A3: If a method passes LNHe, there is no reason to add further dishonest preferences in order to help a former preference. That is, the voter is not being encouraged to add later preferences beyond their honest inclination to do so.

A4: Similarly, if a method fails LNHa, a later preference could harm an earlier one. This acts to discourage voting later preferences.

A2 and A6: see A3 and A4.

Homunq (talk) 03:13, 13 September 2012 (UTC)

0. Re: A3 and A4 (I wanted to see you switch A and B for LNHe, LNHa – but let’s try a different approach because I see where you are coming from and that question is not clear the way I formulated it…)

1. In shortest terms, I don’t think its right to equate Plurality with methods that fail Later-No-Harm by coloring them all red because Plurality cannot fail Later-No-Harm. The colors should be about passing/failing criteria because it’s a compliance table.

2. I think one way to explain my concern with the standard you are suggesting is that it focuses only on the discouraging of later preferences for failing Later-No-Harm. Actually I think the issue is broader than that. It allows the voter to strategically withhold choices to manipulate the voting system. Plurality doesn’t allow this.

3. Note re: failing L-N-Harm: Discouraging later preferences means encouraging “bullet voting” strategy.

4. In Plurality voting, the strategy of bullet voting does not exist, because none of the voters can mark more than one choice. The only way a voter may take advantage of bullet voting is if the voter’s opponents can mark more than one choice. If every voter’s vote is limited to a single mark, then bullet voting is a non-issue. In Range, Approval, Condorcet etc… bullet voting is an issue – i.e. the strategy does exist. These voting systems [rating and some pairwise] are subject to manipulation by bullet voting, but Plurality is not. These are categorically different characteristics of voting systems. By giving them the same color (red) in the LNHa table we are disguising this categorical difference.

5. I can understand how you would want to quite rightfully punish Plurality for not allowing further choices, by assigning color RED. I do see the similarity to how failing LNHa discourages choices; however, in this case, allowing choices while discouraging them creates a vulnerability and is categorically different than not allowing any choices at all, which has no such vulnerability (although not allowing further choices is undesirable for other reasons that are plain). In other words, if we want to rightfully punish Plurality for not allowing further choices, I think this area of the table is not the place to do it – it’s about passing or failing criteria which have to do with vulnerabilities of voting systems.

Specific comments for each Table entry Plurality with LNHa, LNHe…

6. Later-No-Harm/Plurality: Failing LNHa is fundamentally about vulnerability to strategy (i.e. failing a specific criterion), not about whether a multiplicity of choices is permitted on the ballot. In the former case, the vulnerability is considered bad (RED), and in the latter case (i.e. not having further choices), the resulting lack of vulnerability to manipulation is considered good (GREEN). Therefore the Plurality/LNHa cell should be green, not red.

7. Later-No-Help/Plurality: Not allowing further choices is bad on its own, for separate reasons. This shouldn’t change for the method just because we are looking at different criterion. Note: Failing LNHe is bad, not because it encourages adding sincere choices, but for the reason it encourages adding insincere choices and therefore makes the voting system vulnerable to manipulation by insincerely “burying” of an opponent. That’s why it’s bad. That’s why not violating the criterion is good (GREEN). Plurality/LNHe should be green for the reason that it doesn’t violate the criteria which is good, not because it doesn’t allow voter choices, which would still be bad (RED). The standard you are suggesting is the rhetorical equivalent “not having a heart is good because you cannot get heart disease” which I don’t think we want to say. Instead we want to say, separately: "Not having a heart is bad because one cannot live. Not getting heart disease is good."

8. I do understand your reasoning, and I see an argument for it – that not allowing later preferences should be indicated as bad (RED) in the same way that failing Later-No-Harm is bad which discourages choices, but these are categorically different, in my view, based on the reasons above.

9. I would suggest making both cells green, because the colors should not be about distinguishing “encouraging vs. discouraging” polarities, when either one can cause failing of criteria and manipulation, because of how each criterion is defined differently. We need to respect that the criteria are correctly defined for their corresponding subject matter so that pass/fail is meaningful. In other words, the color in the criteria table should be about passing or failing the criteria only, not about how criteria pertain to different aspects of voting systems. Failing criteria is bad (red) and passing or not failing criteria is good (GREEN). Both LNHs are NA, and should have the same color – to me, I see this is a fundamental semantic error in the coloring.

Thanks for listening.

Filingpro (talk) 18:11, 13 September 2012 (UTC)

(undent)OK, I understand your arguments, and they make perfect sense. But I still feel that the LNHa cell should be redder than the LNHe one. Do you agree with this formulation:

➊ LNHa is about whether truncation is a workable strategy. It is not a workable strategy with Plurality, so Plurality technically passes. However, that's because it's mandatory with Plurality. Thus, Plurality has the problems of truncation that one would normally associate with LNHa failure, but not the problems of certain voters gaining a strategic advantage.

➋ LNHe is about whether burial is a workable strategy. It is not a workable strategy with Plurality, so Plurality technically passes. However, that's because bullet voting, which can be seen as a specific, weakest-possible form of burial, is mandatory with plurality. Thus, Plurality has a small part of the problems of burial — mostly, the loss of expressivity — but not the problems of certain voters gaining a strategic advantage.

Certainly if you want to argue that these are both passes, I can't deny that. But I don't think you can deny that (a) they're both in some sense weak or qualified passes, which in this table means some lighter shade of green and (b) ➊ is clearly weaker, and therefore should get a lighter color green, than ➋. So I'm going to try to propose a fix on the page, but I'm open to responses. Homunq (talk) 18:40, 13 September 2012 (UTC)

(note numbers 1,2,3... are for paragraphs and don't correspond explicitly to ➊ ➋ above)

1. OK. Yes I agree that “they're both in some sense weak or qualified passes” because it is categorically different, in my view, when a voting system is simply not applicable at all, as opposed to a voting system which can be applied, and never fails.

2. Thinking aloud – my question is how should this categorical difference be indicated?

(i) Text “NA” distinguishes from “Yes”

(ii) A lighter green distinguishes from darker

3. I would be inclined to rely on the textual difference, because that would reserve the lighter color to mean a voting system that does apply, but fails only under circumstances that require extraordinary assumptions to be true (the latter of which is a highly controversial editorial decision, often qualitative rather than quantitative).

4. In short, I think if we reserve the lighter green for a lower qualitative levels of passing the criteria, then 'NA' should only be distinguished by the text (or some other color?), because 'NA' unequivocally does not fail under any circumstances – so it shouldn’t be confused with systems that partly fail. On the other hand, if we choose uniformly not use lighter greens to mean the latter type of qualitative passing (i.e. partial failures), then we would be free to use the lighter green to distinguish 'NA' from 'Yes'.

5. I see how your ➊ & ➋ above are very well written and incorporate the idea of strategy and make the distinction with the truncation problem and the bullet strategy. My perspective on the truncation problem (which is not the “bullet voting” strategy) is the problem of satisfying the global concept of Majority – that is, Plurality can have a severe vote split and elect a leader with a tiny minority. Likewise, failing Later-No-Harm for multi-choice voting systems discourages saturated ballots and leads to greater chance of Majority failure. So you make a good point. I wonder, though whether this interplay between failure of Later-No-Harm and the problem of satisfying Majority belongs here in the table under LNHa, when it may be that we sacrifice the correct passing/failing of the LNHa criterion itself? Another concern I am having is how we apply this standard uniformly? Evidence of this potential problem may be with the LNHelp explanation – I’m not sure if we can succeed writing the two in parallel. For example, “Plurality has a small part of the problems of burial” - this doesn’t seem true to me but I remain open – so I think we would need to work this out. So I see your point about LNHa failure’s connection to Majority failure and I’m willing to explore this further if it’s important to you, which I suppose could lead to keeping red. My instinct is that it would be preferable to express the interplay between a failure of Later-No-Harm and the problem satisfying of Majority in some other way, because Plurality doesn’t fail Later-No-Harm and red indicates failure.

6. So in summary I would generally suggest at least something green for both. I would caution using the lighter green to qualify 'NA' from 'Yes', IF the lighter green is already used to mean something else. I don’t know if another color introduced for 'NA' would help or harm the table – pardon the pun! Either way, we always have the textual difference “NA” vs “Yes”.

Filingpro (talk) 22:13, 13 September 2012 (UTC)

Less pink

With the recent (and hepful) consolidation of the incompatibility footnotes, I can't see why there should be any pink in the Condorcet or ISDA columns. That should all be red. I suspect if I can't see a reason for the pink, nobody else can, because IIRC I was the one who put it there in the first place (before we had a separate column for strategic majority Condorcet; with that column, the pink seems redundant and even a bit misleading). But I thought I'd ask first here: does anyone disagree? Homunq (talk) 16:24, 14 September 2012 (UTC)

The "pink" in Condorcet and ISDA columns are there because systems which satisfy IIA, Consistency, Participation or LNH can't satisfy the first two due to Arrow's impossibility theorem. Leaving "pink" in IIA/Consistency/Participation/LNH columns but removing them from Condorcet and ISDA columns is biasing the table in favor of Condorcet methods. I would go for a neutral approach. --Wat 20 14:25, 15 Septempter 2012 (UTC)

About IRV and Runoff voting, they both satisfy LNH criterion, which is incompatible with Condorcet and ISDA criteria. --Wat 20 14:25, 15 Septempter 2012 (UTC)

Sorry, I guess I wasn't clear. There are several colors in question. ff7777, which I call "red", is used for flat "no"; while ffbbbb (and in some cases ff9999 or ffdddd), which I call "pink", is used for qualified "no". The reasons for using "pink" instead of "red" vary from column to column. But if I correctly remember the history here, the only reason any system has "pink", and not "red", in the CC and ISDA columns is that, historically, there was not a separate column for strategic, majority Condorcet, and so a footnote in the CC column qualified the failure. Now that there is a separate column, there is no need to keep that "pink"; it can be replaced with "red". Is that clear now? Do you agree? Homunq (talk) 15:25, 15 September 2012 (UTC)

Looking better at the table, all "pink" are related to weaker criteria being met. There is no weaker criteria being met in those columns with the incompatibility references. So, I think it is better to remove all the "pink" related to incompatibility between criteria and leave only the references. --Wat 20 15:45, 15 Septempter 2012 (UTC)

re IRV/Runoff: @Wat20: You're right. I've somehow missed that LNH is mentioned in the footnote, sorry. I just reverted the arguable edit relating to IRV/Runoff.

re incompatibility: I agree on the last point: No "pink", but with references is preferable. --Arno Nymus (talk) 17:15, 15 September 2012 (UTC)

OK, is that good? Homunq (talk) 23:31, 15 September 2012 (UTC)

Since the topic is "less pink", I have to note that Copeland's method is not only vulnerable to crowding, but also to teaming - as shown by the second example for Copeland's violation of IC. Thus, Copeland's entry should not be pink, but red. Since Copeland is the only entry with "crowding", the new comment about other coloring now is somehow "virtual". @Homunq: I would like to leave it to you to advance the table with respect to that fact, thx. Apart from that it's fine. --Arno Nymus (talk) 00:50, 17 September 2012 (UTC)

Good catch. Fixed. Homunq (talk) 02:54, 17 September 2012 (UTC)

Mathematical criteria section looks really ugly.

I want to clean it up by taking the largest groups (absolute/relative/etc) and absorbing them into the criteria descriptions themselves. Right now this is unnecessarily hard to read IMO. Any complaints? 159.1.15.34 (talk) 22:03, 21 December 2012 (UTC)

Whatever happened to the version that said something about "critical failures" ? Helps to distinguish intuitive things like the favorite betrayal criteria from weird, very specific mathematical stuff. Ah well, I've cleaned it up a bit. There's a problem in the bottom of the graph where some cells are crossing columns and they shouldn't. I'll look into it later. 159.1.15.34 (talk) 22:50, 24 December 2012 (UTC)

Well, I personally liked it better the way it was before. But I'm not going to revert because that would violate WP:OWN. What do other people think? Anon IP, are you the same person as Qaanol below? If not, does Qaanol have an opinion on these changes? Homunq (࿓) 12:14, 1 January 2013 (UTC)

Approval/LNH, part 3

Is this method Approval?

~Q1: Is the voting method below Approval?

~Q2: Does it fail Later-No-Harm?

PRESIDENT OF THE UNITED STATES

INSTRUCTIONS:

Mark each candidate that you approve, on the left. Optionally number your approved candidates on the right in order of preference.*

Approve?		Number Ranking (Optional)
[__]	Barack Obama
[__]	Buddy Roemer
[__]	Dennis Kucinich
[__]	Jill Stein
[__]	Mitt Romney
[__]	Rick Santorum
[__]	Ron Paul

*HOW BALLOTS ARE COUNTED:
The candidate who receives the most approval votes wins. Optional rankings do not affect the determination of the winner. (The right column may be used for your benefit.)

Filingpro (talk) 06:06, 11 August 2012 (UTC)

I'd answer yes and no respectively. But the point is, it's debatable, and we don't have RS to back up one side or the other. Homunq (talk) 20:02, 12 August 2012 (UTC)

Note that despite Filingpro's arguments below, this system does NOT fail LNH. An additional candidate can be numbered on the right without approving them. If you reword the system to explicitly forbid that, instead of simply not suggesting it, then that constitutes a significant change, and the resulting system is no longer equivalent to approval. Homunq (talk) 18:48, 23 August 2012 (UTC)

How does a numbered candidate that is not approved cause the voting system to not fail LNH?Filingpro (talk) 01:58, 20 May 2013 (UTC)

Are you arguing the classification of the voting method above is debatable and your position in the debate is that the method is Approval Voting (without rewording) and it does not fail Later-No-Harm? If so, how can you say Approval “does not allow later preferences” when the voter can indeed indicate later preferences between approvals? If later preferences are allowed, then how can Approval not fail Later-No-Harm?

Filingpro (talk) 02:05, 1 September 2012 (UTC)

SIMPLE EXPLANATION: Why all preferential election criteria apply to Approval. Approval fails Later-No-Harm.

Approval Method: Vote for any number of candidates. The candidate with the most votes wins.

Approvals are not equal preferences because the relative preferences between approvals are not specified on the ballot.

Example: Candidates a, b, c, d. Ballot is marked ‘a’ and ‘b’ approved.
a=b>c=d (INCORRECT)
(a?b) >(c?d) (CORRECT) where ‘?’ means a variable (i. e. unknown) preference, either ‘>’, ‘<’, or “=”

Variable preference ‘?’ can be ‘>’ so approvals can be strict preference orderings.

Therefore, all preferential election criteria apply to approval.

Approval fails Later-No-Harm.

NOTE: We cannot say that a preference relation does not exist between approved candidates because then a preference relation could not exist between any approved candidate and a candidate not approved.

Filingpro (talk) 03:09, 22 August 2012 (UTC)

Evidence Woodall applies preferential election criteria to Approval

From Voting Matters Issue 6 http://www.votingmatters.org.uk/issue6/P4.HTM

Woodall: “Majority. If more than half the voters put the same set of candidates (not necessarily in the same order) at the top of their preference listings, then at least one of those candidates should be elected.”

Woodall: “point-scoring systems and approval voting, fail majority.”

~Q: How can Approval fail Majority without considering strict preferences ‘>’ between approvals?

~Q: If a preferential voting criterion applies to Approval, how can it be that other preferential voting criteria do not apply to Approval, such as Later-No-Harm?
Filingpro (talk) 08:35, 23 August 2012 (UTC)

~A1: Precisely because it doesn't consider inter-approval preferences. That is, {A,B,C}>{D} does not have the set {A} at the "top", it has the set {A,B,C}.

~A2: Irrelevant; see A1.

Homunq (talk) 18:38, 23 August 2012 (UTC)

Re: A1...

1. Is it not mathematically impossible to say Approval fails Majority because {A, B, C}>{D} does not have set {A} at the top? In other words, how can Approval fail majority, as you say, if set {A} cannot be considered top? It seems to me, both cannot logically be true. Can you provide at least one example where Approval fails Majority without considering the possibility of relative preferences between the approved candidates?

2. To clarify, I believe the error is when it is assumed that because approval method "doesn't consider inter-approval preferences" therefore no preferences exist, when in fact the preferences are merely an unknown variable “X” which can be either ‘>’, ‘<’, or ‘=’. If our mathematical model is one in which preferences can exist between candidates, we cannot say they do not exist merely when they are not specified on the ballot. The truthful mathematical statement is that these preferences are unknown. We know this because preferences indeed exist between approved and non approved candidates in our mathematical model. We therefore cannot change the mathematical model when we consider the possibility of relations between other candidates. To prove this, note that indeed voters can change the approval cutoff to express preferences between any two candidates.

Filingpro (talk) 01:51, 1 September 2012 (UTC)

And moreover, the page as it stands does not claim that approval passes LNH. It remains neutral (and in fact a careless reader would probably think it says the opposite, as you claim it should). No amount of WP:SYNTH or WP:OR here should change that. Please look for clear statements in new sources, don't just trawl the same old sources for vague hints or rephrase the same old arguments here on the talk page. Homunq (talk) 18:41, 23 August 2012 (UTC)

3. My recent posts set forth a categorically new argument than previous (i.e. Archived) discussions. In prior discussions, both Homunq and Arno Nymus suggested that approvals are “=” preferences. The error lies in this assumption because this is the incorrect meaning (i.e. semantics) of approvals. The correct mathematical model for the preference relation between approvals is not “=” because this information is not obtained by the mechanism of the ballot and therefore the ballot markings do not have this meaning. The correct mathematical model for the preferences between approved candidates is a variable relation “?” where “?” can be either ‘>’, ‘<’, “=”. The arbitrary restriction of “=” will inevitably lead to false conclusions in the field because then we are incorrectly modeling the subject matter at hand - i.e. we are inventing a reality that does not exist. This explains why we have so many absurd results - i.e. Approval and Plurality receive the same Later-No-Harm rating, also Range fails while Approval is not applicable etc. Meanwhile, approval experts assert a voter may optionally cast one of several "sincere" votes by merely adjusting the cutoff. This framework is mathematically impossible if preferences can not exist between approved candidates in the mathematical model.

Filingpro (talk) 01:51, 1 September 2012 (UTC)

Approval/LNH FOOTNOTE WP:OR Problems + Proposal

@Homunq, ArnoNymus and editors:

OVERVIEW: Given the compromise “blank” for Later-No-Harm and Approval, the footnotes cannot contain WP:OR. I suggest the article must contain statements that we all agree are true or a citation is needed.

Footnote for Approval/Later-No-Harm now reads: “Approval and Plurality do not allow later preferences. Technically speaking, this means that they pass the technical definition of the LNH criteria – if later preferences or ratings are impossible, then such preferences cannot help or harm. However, from the perspective of a voter, these systems do not pass these criteria. Approval, in particular, encourages the voter to give the same ballot rating to a candidate who, in another voting system, would get a later rating or ranking. Thus, for approval, the practically meaningful criterion would be not "later-no-harm" but "same-no-harm" - something neither approval nor any other system satisfies”

I will show the problems I see and propose solutions.

PROBLEM #1 “Approval and Plurality do not allow later preferences.”

Q: Do you have a citation for this statement? Otherwise this is WP:OR to me, because it is mathematically inaccurate. Mathematically preference relations do exist between approved candidates, variably as either ‘>’, ‘<’, or ‘=’. You could instead say: “A ballot with only the ability mark approvals does not specify the preferences between approved candidates.” – that would be a true statement I could agree to.

PROBLEM #2 “Technically speaking, this means that they pass the technical definition of the LNH criteria – if later preferences or ratings are impossible, then such preferences cannot help or harm.”

Q: Do you have a citation for this? Otherwise, this is also WP:OR. Again, unknown preferences between approvals do exist mathematically (i.e technically) as either ‘>’, ‘<’, or ‘=’, so the premise of the statement is false. (Please read further for more detailed explanation and solutions).

PROBLEM #3 “Thus, for approval, the practically meaningful criterion would be not "later-no-harm" but "same-no-harm" - something neither approval nor any other system satisfies”

Q: Do you have a citation for this? Otherwise this is clearly WP:OR to me. Again, we can not assume the preferences between approvals are ‘=’ because this information is not specified on the ballot. Any rating method where the number of candidates exceeds the unique possible ratings we obviously can not assume the same rating tells us anything about the relative preferences. (Further discussion below). Furthermore, how can a criterion which no voting system satisfies be practical? It seems to me, the practical criterion would be “Cancellation Immunity” - i.e. “A voter may indicate an additional choice without fully cancelling the probability of each prior listed candidate winning against the added choice.”

NOTE: If we try to explain your position in the controversy in the article more clearly, here are the problems I see that occur:

ATTEMPT #1 (to explain your position in the controversy using NULL preferences): “If it is assumed that because an approval ballot does not specify the relative preferences between approvals that these preferences can not exist, then Later-No-Harm would be considered not applicable to Approval voting because there would be no possibility of relative (i.e. “later”) preferences.”

The error here I see is that the strict approval ballot does not obtain specific preference information between approvals so we can not assume a specific preference relation ‘null’. The only correct mathematical statement is that the preferences between approvals is unknown. (Meanwhile the preference between any approved candidate and an unapproved candidate recorded on the ballot proves that preferences can exist between candidates.)

SIMPLE ANALOGY WHY PREFERENCES BETWEEN APPROVALS CAN NOT BE NULL
If I fail to ask you what your color preference is between red and blue then I can not assume your color preference is the null set - i.e. your preferences between red and blue do not exist. I can only assume your color preference is an unknown variable: either red, blue, equally red and blue, or undecided. (Meanwhile, if I also ask you what your color preference is between blue and yellow and you give a preference, then I have proof that preferences can exist between colors.)

ATTEMPT #2 (to explain your position in the controversy using ‘=’ preferences): “If it is assumed that the relative preferences between approvals must be equal preferences, and if Later-No-Harm applies only to strict preference orderings (rather than “equal” preferences), then Later-No-Harm would be considered non-applicable.”

Again, the error here is that the strict approval ballot does not obtain preference information between approvals so we can not assume a specific preference relation ‘=’. If we falsely conclude that these preference must be ‘=’ just because the counting method treats them this way, then a strict preference voting method which counts all ranked candidates as equal would not be a preferential election rule but it is, which leads to a contradiction.

In summary, approval ballot markings do not say anything about the preferences between approved candidates. If A and B are approved on a ballot this does not tell us anything about the preference relation between the two. The mechanism of the ballot does not capture this information. Therefore, we can not falsely assume preferences between approvals are always ‘=’ or always ‘null’. The only correct mathematical statement is that the preferences between approvals is an unknown variable, either ‘>’, ‘<’, ‘=’, or ‘?’.

For example approval of [AB] for candidates A,B,C. The mechanism of the ballot records: A>C, B>C. No further specific preference information can be assumed. This would clearly be a logical (i.e. mathematical) error to do so.

The problem when we try to explain the controversy from our talk page is that it sounds like something that belongs in a talk page, not in an encyclopedia article. We also risk putting mathematically incorrect statements in the article. Is there a citation for the controversy itself?

PROPOSAL: We could make statements we all agree are true or have a citation.

For example,

“A citation for the explicit determination of Later-No-Harm’s application to approval has not been identified. A voter changing an approval for candidate 'A' to the approval of both candidates 'A' and 'B' harms candidate 'A' according to the definition of local and relative criteria. The definition of Later-No-Harm as a local and relative criteria was proposed by Woodall, Douglas (December 1994), Properties of Preferential Election Rules,Voting Matters (3). Other preferential election criteria such as Monotonicity and Majority are commonly applied to Approval voting, as in Woodall, Douglas (December 1996), Properties of Preferential Election Rules, Voting Matters (6).”

Can this proposal be improved upon? Thank you.

Filingpro (talk) 07:33, 4 September 2012 (UTC)

Let me try to take a step back here, if I may. We agree that the article should follow the sources. We'd also both like it to take what we see as obvious steps to clarify the meaning of those sources, but since we disagree on the correct interpretation of LNH in this context, we're disagreeing on what those steps should be, and we both see the suggestions of the other side as WP:SYNTH that is leaning towards an incorrect conclusion.

It seems to me that this is exactly the situation which rules like WP:SYNTH are made to cover, and do cover well. The current footnote, indeed, goes too far (in my direction). But I think some of Filingpro's suggestions go too far in the other direction. For instance, their* use of the Woodall citation in their final proposal above seems to me a clear case of WP:SYNTH (and, in my opinion, an incorrect one; but this isn't the forum for that). *neuter "they", sorry I don't know your gender; feel free to edit to your preferred pronoun and delete or strikeout this note.

So I think we have to dramatically scale back the footnote. Unfortunately, that probably will leave it where a naive reader won't even fully understand Filingpro's or my positions (both of which are probably shared by other well-informed people). But then again, this talk page amply documents both sides.

What CAN we agree on for the footnote? I'd say that Filingpro's suggestion “A ballot with only the ability to mark approvals does not specify the preferences between approved candidates.” is an acceptable starting point. Augment this with a statement that there's disagreement about approval/LNH, and I think that's as far as we can go. Homunq (talk) 23:20, 4 September 2012 (UTC)

I just want to indicate that the current footnote is already a compromise. It contains sentences for both "views". If Filingpro challenges the sentences that tend into one direction, I have to state that there are also statements, that tend in the other direction, I accepted only as part of the compromise formulation, which I would obviously not accept, if Filingpro strikes the other statements. E.g. "However, from the perspective of a voter, these systems do not pass these criteria." is only acceptable in the combination with "Technically speaking, this means that they pass the technical definition of the LNH criteria".

However, I think Homunq is right, that the footnote therefore have to be rewritten from the scratch. I will think about it and 'll try to add some helpful input after that. --Arno Nymus (talk) 16:09, 5 September 2012 (UTC)

Thank you Homunq and Arno Nymus for listening to my concerns and I am open to suggestions you make. I believe I do understand the points each of you are making. Filingpro (talk) 22:11, 5 September 2012 (UTC)

Q: Is this posting to everyone's liking?

“A citation for the explicit determination of Later-No-Harm’s application to approval has not been identified. A voter changing an approval for candidate 'A' to the approval of both candidates 'A' and 'B' harms candidate 'A'. Meanwhile, a ballot with only the ability to mark approvals does not specify immutably the relative preference between any two approved candidates. The definition of Later-No-Harm as a local and relative criteria was proposed by Woodall, Douglas (December 1994), Properties of Preferential Election Rules,Voting Matters (3). Other preferential election criteria such as Monotonicity and Majority are commonly applied to Approval voting, as in Woodall, Douglas (December 1996), Properties of Preferential Election Rules, Voting Matters (6).”

NOTE: We can not say for approval A the addition of approval B [AB] does not harm 'A' because the definition of harm does not rely on the word "later".

Filingpro (talk) 16:34, 6 September 2012 (UTC)

No, that is absolutely unacceptable. As I already said, it uses WP:SYNTH (which in my opinion is also a serious misinterpretation, or at least over-interpretation, of Woodall) to present just one side of the issue. Homunq (talk) 16:42, 6 September 2012 (UTC)

Q: How is the following WP:SYNTH "A voter changing an approval for candidate 'A' to the approval of both candidates 'A' and 'B' harms candidate 'A'"?

Filingpro (talk) 19:53, 6 September 2012 (UTC)

Q: Isn't this a true statement we can all agree on? My intent was to agree to adding a true statement also about your position (see below) as a compromise, without getting into our talk page debate which in my opinion doesn't belong in the encyclopedia article. To respect our compromise, we would have to say things like "if it is assumed that..." then some SYNTH logic after that.

Is something like below better...what do you suggest?

"A voter changing an approval for candidate 'A' to the approval of both candidates 'A' and 'B' harms candidate 'A'. On the other hand, a ballot with the ability only to mark approvals does not specify the later preferences between approvals."

To me this seems simple and clear and makes true statements I hoped we can all agree to. For each of us, we consider the other statement to be IRRELEVANT to the issue, but at least they are true statements and this seems a reasonable compromise.

Filingpro (talk) 20:28, 6 September 2012 (UTC)

The following summary of our talk page debate is offered as an illustration, something like...

"If it is assumed that approvals can be considered preferences in any order, then Approval fails Later-No-Harm. On the other hand, if it is assumed that approvals must be equal preferences, and if we assume Later-No-Harm can only be applied to strict orderings and not equal preferences, then Later-No-Harm is not applicable to Approval."

Filingpro (talk) 20:51, 6 September 2012 (UTC)

Arno Nymus states the current footnote is a compromise. ~~No. This is incorrect.~~ I see this differently. The disagreement is over the math. The footnote asserts a false (in my opinion) mathematical position to be true, and then says something about the voters perspective. That is not my position. My position is that your math is ~~dead~~ wrong and needs a citation (even though its obviously false, in my opinion). When you quantize something that is infinite, you can't say the same quantum values mean anything about the infinite. This is a mathematical axiom you are violating. Please think about digital music. Two of the same numbers don't mean equal points in the sound wave. It just means your sample size is limited, not infinite. This is a mathematical axiom of quantization. I believe you are failing to understand the mechanism of a rating ballot with respect to preferences. The analogy is exactly the same. In this way, you are constructing a false mathematical model of the ballot. This does not belong anywhere in the article in my assessment.
Filingpro (talk) 21:10, 6 September 2012 (UTC)

I, in turn, think that you are simply, provably wrong on the math. Unlike you, though, I don't see any benefit to arguing it out on the talk page.

Your proposal is unacceptable. Would you like to propose another wording that (a) states that there is disagreement and (b) indicates that the source of the disagreement is something about the lack of a 1-1 mapping between a preference order and an approval ballot? If you can find a (c) we'd agree on (after talking here) then that's great too but prospects look dim.

Go ahead and try a bold edit on the page itself as long as you don't go beyond (a) and (b). Homunq (talk) 21:41, 6 September 2012 (UTC)

1. Before I make any edits, can you please first edit the table to separate Plurality from Approval in the LNH footnotes? You have done more edits on this table than I have, I believe, and you obviously have a system for the columns etc. I don't want to make an edit for Approval and have it reverted for some trivial technicality and especially a problem with intersecting Plurality. For the time being only, I don't care what you do with Plurlity LNH - we'll deal with that later if necessary. In the meantime, having the two together in the LNH table is a mathematical abomination ~~by any standard~~ in my opinion. Plurality and Approval are categorically different voting systems. In a strict approval ballot there can be a multiplicity of candidates marked, whereas in a strict Plurality ballot there can be no more than one candidate marked. If you acknowledge and respect our compromise, then for Approval there is debate over how the multiplicity of approvals can not be mapped 1-1 and what this means, whereas in Plurality there can be no such debate because there is no multiplicity of candidates. Having said that there is no question to me that the Plurality LNH should be green and NA. Please let me know when you are done with this edit and then I will proceed.

2. re: (a) your request for stating disagreement above, how about what I said before: "A citation for the explicit determination of Later-No-Harm’s application to approval has not been identified."

3. re: (b) your request to indicate something about lack of 1-1 mapping between preference order and approval ballot, before I post, do you agree there is a lack of 1-1 mapping for the preference order between any two approved candidates, and do you agree there is an unequivocal 1-1 mapping for any approved candidate 'A' and any unapproved candidate 'B' to A>B and visa versa?

Filingpro (talk) 16:03, 7 September 2012 (UTC)

1. We're working for consensus here. I'm not going to revert whatever you do, I'm going to work to improve it. That said, I'll get to this later, if you don't beat me to it.

2. That's an acceptable start, though I'd like to phrase it in a way that doesn't strike such a jangly note of wikipedia process trivia.

3. Yes. But lets try to avoid being that specific. There's not a 1-1 mapping overall; since we don't have citations, any further details could easily start the slippery slope to WP:SYNTH.

Homunq (talk) 17:47, 7 September 2012 (UTC)

1. Thanks.

2. Good point – I agree. ~~Does “published source” instead of “citation” help?~~ The best solution may be not to include such a statement because it is already self evident that a citation has not been identified or a disagreement exists because of the blank or "?" in the table - I think anything we say here will expose wiki process or cite a disagreement on our talk page which probably doesn't belong in the article.

3. Do you agree if we make a single statement (re: lack of 1-1 mapping), the most mathematically precise and accurate is:

“There exists no 1-1 mapping for any two approved candidates to a specific preference ordering.”

I believe this is the only relevant statement because the debate is exclusively over what the ordinal preferences are between approvals, and whether or not they can be considered “later”. Am I correct?

Any less precise (i.e. more broad) statement, in my assessment, would objectively prove editorial bias because it would only serve to hide information about very subject matter at hand (by artificially broadening the subject to obscure relevant details about the actual subject), while only being equal with regard to WP:SYNTH (both require the same presumption to be taken as given, that there is a lack of 1-1 mapping for any two approvals to a specific preference order.)

Filingpro (talk) 22:03, 8 September 2012 (UTC)

re (3): No, because there's also no 1-1 mapping between any two unapproved candidates, or for that matter between any 3 or more approved or unapproved candidates. That's why I'd prefer something like (working from your wording): "There exists no 1-1 mapping between a specific approval ballot and a specific preference ordering." Sound good to you, or do you have a counterproposal?

Slowly making progress, Homunq (talk) 15:19, 10 September 2012 (UTC)

1. I agree we are making progress and I believe I understand your point of view. I like the wording of your broad statement. The problem I see is anything we say about lack of 1-1 mappings is the same level of WP:SYNTH. To make the broad statement you suggest requires presuming either approvals or unapprovals cannot be 1-1 mapped. Do you agree? We therefore cannot say the broad statement is any less of WP:SYNTH than a statement which simply points out specifically the lack of 1-1 mapping for approvals. (Incidentally, I don’t mind if we say “any two approvals” or something about “amongst approvals” - that's fine - I understand your other point re: the 3 approvals).

2.So which do we choose? The specific statement reveals that approvals cannot be 1-1 mapped (the central issue in the debate), while the broad statement hides this information. I would ask, why do we want to hide this?

3. Perhaps more importantly, and perhaps more of a concern re: WP:SYNTH is what statements will follow as a result of selectively choosing the given presumptions about the 1-1 mappings (or lack of). (Note also: if given 'x' has 1-1 mapping and 'y' does not, then the statement "'x' has the only 1-1 mapping among 'x' and 'y'" is not WP:SYNTH because it only restates what is given, AND is more accurate than "at least one of 'x' and 'y' is a 1-1 mapping" which hides information. Likewise, in a discussion about 'y' we obscure the truth by choosing the latter statement.)

4. Whether or not we agree right now, perhaps it would be helpful to start by coming up with a reasonably objective statement of the two opposing positions we can agree on (I believe this may also inform us with regard to what to say about 1-1 mappings or make it less of a contentious issue?)

5. Here is my best attempt at summarizing…can you improve the summary statement of the mathematical disagreement?

6. "If it is assumed that the relative preferences between approvals are unknown (and therefore can be considered in any order), then Approval fails Later-No-Harm. On the other hand, if it is assumed that the relative preferences between approvals must be equal (and therefore cannot be considered later preferences), then Later-No-Harm is not applicable to Approval."

Thanks.

Filingpro (talk) 22:06, 10 September 2012 (UTC)

I don't think that's even true, and it's precisely because you're allowing preferences among approved candidates but ignoring preferences among unapproved candidates. If I prefer A>B>C and A is winning, I could harm A by voting A>B(>C). But those preferences are also perfectly consistent with A(>B>C). The latter vote does not harm A, of course.

Yes, that's pretty much pedantry when it comes to the spirit of LNH; the latter ballot doesn't meaningfully "include" the B>C preference. But the fact is, it includes it precisely to the same degree that the former includes the A>B preference. An approval supporter could claim that the pedantry is the idea, embedded in the definition of LNH, that any preference profile translates into one and only one "honest" ballot; and that the criterion's failure to mathematically give the intuitive result is just a reflection of the original fault in its definition. Homunq (talk) 00:25, 11 September 2012 (UTC)

1. @Homunq, sorry I'm confused about what you are referring to above, so I added numbers (1,2, 3...) to my previous post. Can you please use the numbers I added to show me what parts you are responding to? Thanks.Filingpro (talk) 03:29, 11 September 2012 (UTC)

2. I'm not sure what you mean when you say I'm "ignoring preferences among unapproved candidates". In my mathematical model, for candidates A,B,C,D when [AB] are approved I model this as [A?B]>[C?D], where ? is an unknown preference relation that can be any of '>', '<', or '='. So I am treating the preferences among approvals the same as among un-approvals. I don't understand your point?

Filingpro (talk) 04:14, 11 September 2012 (UTC)

3. Note: re: 2. This is the only mathematical model to correctly translate between same ratings and ordinal preferences; however, LNH allows for truncation of preference listings which is inherent to its definition, so unapproved candidates are not considered listed on the ballot (they are truncated). So in the example I gave above, D is not considered added to the ballot to harm C already listed, because C is not listed, and D is not added to the ballot.

Filingpro (talk) 04:38, 11 September 2012 (UTC)

4. Another example: In Plurality voting with A, B, C, a vote for A is mathematically A>[B?C]. Note: B and C do not harm each other because they are not marked on the ballot (i.e. not added to the listing). LNH definition is crystal clear that a candidate must be marked on the ballot to be considered to cause harm. This also is obvious and intuitive from a voter perspective, of course.

Filingpro (talk) 05:13, 11 September 2012 (UTC)

5. @Homunq – my intention was to compromise by simply stating our mathematical disagreement in a neutral way that doesn’t assert either to be truth. I thought the disagreement was that I consider the ordinal preferences between approvals to be unknown while you and Arno Nymus consider them to be equal. Is this not correct? Please explain. I am trying to be helpful and cooperative and make this simple and easy. Can you help with this?

Filingpro (talk) 06:00, 11 September 2012 (UTC)

The part of the following comment that helps leading to a result is only the last point; however, since Filingpro asked for a point of view, the other points deal with that.

Your ?-model is WP:OR and I don't think that it is accurate to describe the situation. How would your model deal with three approved candidates?
Maybe instead the model of the "unknown order"-operator '[]' would be helpful for your idea. This operator says, that all candidates within it are in an unknown, but valid preference order, so "A > [B C]" means "A > B > C" XOR "A > C > B" XOR "A > B = C". Every approval ballot would have the form "[A B C ...] > [Z X Y ...]". However, I obviously made this operator up right here and thus, it is WP:OR, too, and so I will not go into details.
ad (2012-09-10, 22:06, 4.): At this time, I don't see that this would be helpful for me, but if it helps you, here is my point of view in two parts (and two notes):
The criterion states "Adding a later preference to a ballot should not harm any candidate already listed." So, it has a premise and a conclusion. The premise is "Adding a later preference to a ballot". The premise says, that for a violation case it is necessary to add a "later preference to a ballot". This means, that a candidate have to be marked as later on the ballot. The only possibility to mark a candidate B as "later" (against another candidate A) on an approval ballot is to mark him as unapproved (as long as the other candidate A is approved). This can never harm A. Thus, Approval cannot fail LNH.
However, since this addition of a later preference on the ballot is effectively limited to one for Approval ballots, in nearly every case, even the premise can't be true. So, this satisfication of the criterion is technically and mathematically correct, but means very little. And in fact, Approval ballots are just something different than preferential ballots and there is no mapping* from the set of approvals to a valid preference list, so in his "spirit" the criterion can't be applied.
Note, that the lack of a mapping is not necessarily a problem for a criterion that doesn't explicitely demand an action on the ballot. But, obviously, this actual action on the ballot, is the fundamental core of the later-no-harm criterion.
* In fact, I think, it is sufficient to say that there is no mapping in this direction, since we have approval ballots and need a preference list. So, the other direction would not be necessary and therefore, there is also no need of an bijective ("1-1") mapping. --Arno Nymus (talk) 13:01, 11 September 2012 (UTC)

1. Yes I believe my model and your model are equally WP:OR/SYNTH. I believe the question what we do in the article or whether we can perhaps come to agreement on the math, or at least agree on what we are disagreeing about? Mathematically, in my model, three approved candidates [ABC] are modeled [A?B, B?C, A?C]>(other candidates). There may be a better notation for notating an unknown preference order between three or more candidates but I believe this conveys the concept clearly.

2. Yes – I see that both of our mathematical arguments are WP:OR currently without citation. Yes I like the notation you suggest. Much better. Thank you.

4. Yes I understand your point of view. Where we can hopefully agree to disagree is the sentence: “This means, that a candidate [has] to be marked as later on the ballot.” We see this differently. I believe that your interpretation is literal and mine is mathematical, and that the mathematical interpretation is correct. Your including the word "as" rhetorically makes the requirement literal, which I believe is incorrect. Perhaps I can persuade you or vice versa. Here is what I see, in my view, mathematically…

4a. First, as you agree, ratings are a different mathematical model than ordinal preferences.

4b. A logical (i.e. mathematical) translation exists, in my view, which is not always a 1-1 mapping, as we have seen. I believe when A:1, B:0 we take this axiomatic to mean A>B. Based on my analysis, A:1 B:1 does not mean ordinal preference ‘=’, it means mathematically unknown ordinal preference. To me, this is the key concept to understanding how to correctly and meaningfully relate the mathematical model of ratings to ordinal preferences. If I ask you to rate four professors A, B, C, D on a scale using “Good, Ok, Bad” you might say A and B are “Good”. This doesn’t mean you do not have a preference between the two because I never asked you for that information (i.e. you may prefer A>B or B<A. Clearly, in my view, it would be a logical error to assume you cannot).

4c. The mechanism of the strict Approval ballot and Range voting do not capture ordinal preferences. They capture ratings. Therefore, based on 4b above, the marking of the approval [AB] mathematically means one of three ordinal preferences can exist: either A > B, A < B, or A=B (or add A?B, if you like). So when A and B are approved on the ballot, yes indeed it is possible that a later preference has been marked on the ballot. This is my mathematical model. I understand you have a different model. I am looking mathematically at what an approval marking is on the ballot – i.e what it means mathematically (not literally) with regard to ordinal preferences. I believe if I understand your model correctly, in my view you are correctly taking A:1, B:0 to mean later ordinal preference but then you are incorrectly assuming A:1, B:1 to mean immutably ‘=’ ordinal preference, instead of unknown. It’s my current understanding, in simple terms, that you maybe mistaking the literal for the mathematical. As you know, in math we can redefine literals to have different mathematical meanings. The literal markings A:1 B:1 only mean same rating. The mathematical meaning for ordinal preferences is unknown.

4S. (summary) To summarize, the candidate does not have to be marked literally later on the ballot. Two candidates can be given the same rating on a rating ballot which mathematically means an unknown ordinal preference exists and therefore either marked candidate can be a later preference. The literal interpretation is not a mathematical interpretation, in my view.

5a. re: “limited to one” – what about candidates A B C D with approval [AB] – doesn’t this mean A>C, A>D, B>C, B>D, so you have more than one later preference? In any case, I don’t think this was your central point.

5b. re: “there is no mapping* from the set of approvals to a valid preference list”. I would say there is no guaranteed 1-1 mapping; however, this doesn’t mean there is not a logical mathematical meaning for ordinal preferences based on approval ratings, using unknown variables which are common to math – for example, suppose X is a variable that can be either {1, 2, 3} then X can be 3. Note also Range voting is applied to Later-No-Harm without controversy so I don’t believe we can make a complete statement, at least consistently, that ratings cannot be applied to ordinal preference criteria. Also Majority is applied to Approval by Woodall himself. (Voting Matters issue 6).

7. I think I see your point, although if its sufficient to say there is no mapping in the direction of marked approvals to ordinal preferences among them, then isn’t the only correct mathematical statement that the preference order between approvals is an unknown variable in our model?

Filingpro (talk) 00:17, 12 September 2012 (UTC)

Now that the article is fixed, how does any of this relate to the article? Homunq (talk) 00:29, 12 September 2012 (UTC)

re 1: "[A?B, B?C, A?C]" does not exclude cycles.

re 4: I disagree with your usage of the term "mathematical". Whereas you're right that I take Woodalls literal definition "literal". Since we only have a literal definition* we have to deal with it. So taking it literal is the only way to stay out of WP:OR; since every other approach (than taking a given definition literal) is an (free) interpretation. Because of WP:OR we cannot speculate that Woodall, if asked explicitly for Approval, would have changed his wording, so we have to deal with the formulation that is actually in existence.

This should not be a play on words, but sadly we do not have a formal definition, but a literal one and with literal definition there are essentially two ways: Interpreting it or taking it "literal".

re 4b: Mathematically, I think you want to say, that there is no canonical mapping, but there is a canonical relation. And with that relation, your interpretation of LNH is violated by approval voting. However, still if you take the definition literal and taking this relation in account, approval still mathematically satisfies LNH. As can be seen, "literal" and "mathematical" are NOT two mutually excluding types of points of view. In fact, we have to use a point of view that is both, "literal" for not violating WP:OR and "mathmatical" for being accurate.

re 5a: Right, "effectively limited to one" stands for something like "only one step". Since I expected this part to be uncontroversial, I passed to define this exactly.

re 5b: "I would say there is no guaranteed 1-1 mapping" I take that as an affirmation of my proposition; since obviously, my statement uses "mapping" in its mathematical meaning in the context of the given entities of voting theory. If you extend entities (in this case WP:OR), you can falsify nearly every statement.

"Majority" does not need any marking of preferences on the ballot, so it is a different situation. For, Range voting, it is easy to create an example with strict different ratings between two candidates that translate unanimously into a strict preference order between them on one ballot with the lower candidate winning because of not being dumped to 0 points. With approval voting, no such violation case for LNH is possible.

However, Homunq is right. So, let's agree that we have different views on that point and hopefully, that we have a text version for the article, we all can live with and close this topic with this pleasant result. Best wishes, --Arno Nymus (talk) 19:46, 12 September 2012 (UTC)

AXIOM: Same limited ratings can not be assumed equal continuous preferences

A continuous straight line (i.e. preferences) using limited ratings

AXIOM: Same ratings (red) can not be assumed equal continuous preferences.

Approvals are intrinsically ratings (limited).

Preferences are intrinsically continuous (i.e. infinitely small differences between ratings for A > B can exist).

What is intrinsic to approvals and preferences can not be exempt from the axiom. The axiom does not require a citation. Approval fails Later-No-Harm.

Filingpro (talk) 03:37, 6 September 2012 (UTC)

"Preferences are intrinsically continuous (i.e. infinitely small differences between ratings for A > B can exist)." No. (Continous) Ratings are intrinsically continuous. Preferences are by definition ordinal and thus, not necessarily directly connected to concrete values on a rating, but - at most - singly the relation that says whos value is bigger. (mathematically in short: "An order relation is not (at all) the same as the real number ray")

Since the assumption does not hold, the "axiom" doesn't even make sense. Best wishes, --Arno Nymus (talk) 12:06, 6 September 2012 (UTC)

I thought we were moving towards a resolution. It's really not helpful for Filingpro to go back to pushing WP:OR. Really, we get the logic of what you're trying to say, but it isn't an AXIOM we accept, and more importantly, it doesn't come from a WP:RS. Let's trim the footnote to something we can agree the sources support, and stop this pointless arguing. Homunq (talk) 12:20, 6 September 2012 (UTC)

re: Arno Nymus response
1. If we translate A>B>C using the ordinal preference model to a mathematical model where only limited ratings of 0 or 1 can exist, clearly no definitive 1-1 mapping exists for translating both A>B, and B>C simultaneously. Mathematically at least one of these preference relations will be lost, because it must be the case that either A and B receive the same rating or B and C receive the same rating. Therefore, we cannot assume that and same rating translates to equal ordinal preferences because A and B having the same rating can clearly mean A>B. QED.
2. Note: the mechanism of the strict approval ballot does not capture ordinal preferences, it captures limited ratings. We must translate to the other mathematical model to say anything about ordinal preferences. If we falsely assume same limited ratings are equal ordinal preferences clearly we have made a gross mathematical error.
3. @Arno Nymus. For the record, are you claiming the following is not axiomatic OR true?... “Same limited ratings cannot be mathematically assumed equal ordinal preferences.”
Filingpro (talk) 22:33, 6 September 2012 (UTC)

I'm not Arno, but what I'd say is that it's not grammatical OR germane.

Please stop arguing the substance of this issue; our job here is to find an acceptable way to treat this issue on the page. You have a valid point: the page as it stands veers into WP:SYNTH, and needs fixing. But arguing about axioms here is doing nothing to help that. Homunq (talk) 23:11, 6 September 2012 (UTC)

New Footnote For Approval LNH

The new posting attempts to reflect the compromise (see other postings) and avoid asserting anything to be unequivocally true that is mathematically under debate and has no citation. General remarks were added to keep the spirit of the recent posting, while keeping to statements that have generally been agreed to be true by editors in prior discussions. There is still some ongoing debate.

The argument for NA/Pass has been asserted to be mathematical. An opposing mathematical position has emerged. Neither has an explicit citation nor have editors been persuaded.

The two conflicting mathematical positions seem to be (1) that approvals mathematically are unknown ordinal preferences and therefore can have later preferences, and (2) later preferences do not exist mathematically for approvals.

To respect the compromise until a citation found it’s my request that neither of the two mathematical positions be asserted as truth in the article, which the prior posting did. There seemed to be some general agreement that we had a WP:OR problem; hence this update.

This issue has been very contentious and nuanced so please consider getting some background before making edits/reverts (see other postings on the talk page). I hope improvements can be made that still reflect this compromise and I invite others to assist. Thank you.
Filingpro (talk) 08:26, 11 September 2012 (UTC)

I have to say that I find it really dizzying that you start so many sections for that one topic. I can't even imagine, that anyone that has not been part of the discussion from the beginning, could have the possibility to follow the topic, if it spreads irregularly into new topics. Therefore, I changed the sections to subsections.
I modified your version, mostly small wording changes and the addition that LNH is about marking something on a ballot. But, I also had to remove one sentence, that I disagree in this formulation: "If it is assumed that the relative preferences are mathematically unknown (and may exist in any order), then Approval fails Later-No-Harm and passes Later-No-Help". I disagree as is implied by the statement of my point of view above. I also out-commented the corresonding "counter-sentence" that LNH is not-applicable in the counter case of the assumption. The formulation of that part have definitely to be reworked.
However, I think the first part of your approach is accurate (with slight rewording):

"The definition of Later-No-Harm and Later-No-Help requires the marking of a later preference on the ballot that helps or harms an earlier preference. On an Approval ballot only approvals are specified and no (i.e. later) preferences between approvals (although these later preferences may exist from a voter’s perspective)."

--Arno Nymus (talk) 13:34, 11 September 2012 (UTC)

Thank you, Arno and Filingpro. From my perspective, the current state of the page essentially resolves this long-burning issue. Now, that wasn't that hard, was it? :) Homunq (talk) 15:42, 11 September 2012 (UTC)

@Homunq – I do appreciate your good humor here. It can be challenging to keep positive and build camaraderie in an online debate, especially a controversial issue amongst well meaning people. At least we are improving and modifying the listing. I think that's a good sign. Thanks.Filingpro (talk) 06:31, 12 September 2012 (UTC)

I modified the posting for the following reasons (NOTE: the #numbers correspond to the sentences – not responding to prior comments)

1. Re:“The definition of Later-No-Harm and Later-No-Help requires the marking of a later preference on the ballot that helps or harms an earlier preference.”

1a. I understand you wanting to say something about the ballot because your viewpoint is based on literal markings of ratings and you want to apply that in a certain way to a criterion about ordinals. I can cooperate with your request, however, the problem with adding “the marking of” and “on the ballot” alone to the definition is that Woodall defines the ballot to be a strict preference ballot, yet in this context the reader must assume we are talking about an Approval ballot. When we speak of the definition of Woodall’s LNH we must say what ballot he is talking about, whereas this sentence (especially followed by your next sentence) implies Woodall includes strict Approval ballots in his definition, but he does not. The new sentence I added correctly represents the definition from Woodall without taking “ballot” out of context to mean something other than what it means in Woodall’s framework.

1b. NOTE: (re: 1a) The problem in general as I see it is that Later-No-Harm does not require the literal or explicit marking of a later preference when the ballot is an Approval (or rating) ballot, because any two marked approvals mathematically can be later preferences. I believe you are confusing ratings with ordinal preferences (or not translating properly). The approval ballot is a rating ballot. Later-No-Harm is defined for ordinal preferences. Two same ratings on the ballot are unknown ordinal preferences. Again, I believe you are assuming the literal markings of ratings have the wrong meaning for ordinal preferences.

2a. Your second sentence I generally liked although we lost what (i.e. later) was referring to and was missing “not” grammatically. I fixed that to say the same thing now clearer.

2b. I also changed the parenthetical remark regarding existence of preferences between approvals to “(although preferences may exist, if only from a voter’s perspective)”. Why? The prior listing rhetorically implied that the later preferences do not exist but only from the perspective of the voter. This doesn’t respect our compromise over the math. The compromise is that later preferences may or may not exist mathematically – we must treat each view equally. The new wording matches that requirement – i.e. has that meaning. If you don’t like this you could take out that entire parenthesis comment. Is that fair?

3 I took out the rhetorically repetitive “from a voter perspective”. Also mathematically, each additional approved candidate either harms the probability of the other approvals or it doesn’t. It doesn’t matter what the voter perspective is. This statement also has nothing to do with “later” so there should be no controversy here. Filingpro (talk) 06:31, 12 September 2012 (UTC)

I made a couple word edits 1 2.Filingpro (talk) 06:43, 12 September 2012 (UTC)

new version looks fine to me. Homunq (talk) 11:09, 12 September 2012 (UTC)

re 1a: What do you mean by a "strict" approval ballot? What would be the difference between an "approval ballot" and a "strict approval ballot"? Best wishes, --Arno Nymus (talk) 18:37, 12 September 2012 (UTC)

Thanks. Good point or good question. I think regardless of my answer, it points out that we don't explicitly define the difference in the article, and it would be WP:OR or veer into WP:SYNTH to use my example ballots. So I have no problem if you would like to remove the word unless its clarified somehow without WP:OR (see my edit below). In my view, many ballots technically exist that are not strictly limited to approval ratings and are Approval ballots - these are ones where a voter can optionally mark ratings, or is simply asked to write down approvals in order of preference, or the ballot has a fixed, arbitrary preference ordering invisible to the voter. I understand you would categorize these ballots as not Approval ballots. That's ok. To me, the last encoded ballot I mentioned is an Approval ballot because the voter is completely unaware of marking anything other than approvals and is only permitted to mark approvals. Again, I am ok with you removing the word "strict" because it has not been clarified you make a good point - "strict" is vague/unclear.

Filingpro (talk) 20:49, 12 September 2012 (UTC)

MINOR edits after second read today + Arno Nymus good comment. # numbers below correspond to sentences (not prior posting on the talk page)

1. Smoother read, same meaning “a strict preference ordered ballot” changed to “a strictly ordered preference ballot”

2a.”strict” was vague, so we now say ~~“with only strictly limited ratings”~~ "strictly with limited ratings" - edit is perhaps less wordy and equally clear.

2b.Smoother read, same meaning – I removed the repetition of “records” then “does not record” – more concise.

2c.I made a semantic error on the last posting. “if only” still implies only from a voter perspective and the statement was vague/unclear. I corrected this by reverting to the previous parenthetical statement approved by all editors, but merely changed the word “although” to “while”. “Although” also rhetorically implies only from a voter perspective. “While” is neutral because it merely states the fact as concurrent rather than implying any relation to the prior remark that veers further into WP:SYNTH. If this neutral, tiny change is strongly disagreeable to anyone, you could still remove the entire parenthetical statement. I hope this is agreeable to all. Thanks.

Filingpro (talk) 20:49, 12 September 2012 (UTC)

As you offered above for the "strict", I would like to remove the additional ", strictly with limited ratings,", so the sentence would just be "An Approval ballot records approvals but does not record explicit relative (e.g. later) preferences between approvals (while preferences exist from a voter’s perspective)." Mathematically, an approval ballot is just the set of candidates approved by one voter or - by a well known mathematical fact - equivalently a map from the set of candidates into the bool set (with true = approved). Also, the second part of the sentence explains one more time what information an approval ballot contains and what information it does not cover. So, I think, the insertion about ratings is more likely to confuse the reader. He could e.g. think, that this is a very special case of approval ballot, and that this footnote only holds for these special cases, so he would think '"normal approval" satifies LNH'. Best wishes, --Arno Nymus (talk) 15:26, 13 September 2012 (UTC)

Sure. You can do the honors of removing the phrase. It would also make the whole thing less wordy. I concede the paragraph is not exactly poetic, but gets the job done. I agree with your mathematical model of approvals as ratings, and I also agree from a voters perspective your model is the starting point in the chain, certainly - the essence of the approval concept. Understanding Approval with respect to ordinal criteria requires translating the model, as you say, a relation exists. In other words, Approval does have a correct mathematical model in ordinal terms [A1, A2, A3...]>[a1, a2, a3...] where '[...]' means unknown preference ordering, while big 'A' means approved and little 'a' means unapproved. This is why, in my view, Approval is a preferential election rule. Alas, more theory...Regards,

Filingpro (talk) 19:52, 13 September 2012 (UTC)

We might find something useful in Gibbard's original 1973 proof of the Gibbard-Satterthwaite theorem (Gibbard, Allan. “Manipulation of Voting Schemes: A General Result.” Econometrica 41, no. 4 (July 1, 1973): 587–601.). For instance on pp 590-591: "Any non-chance system of decision making is characterized by a game form in a clear-cut way, but manipulability is not a property of game forms alone. It is rather a property of a game form plus the functions σ1,,. .., σn which characterize honest voting. Equivalently, it is a property of the voting scheme defined from the game form g and σ1, . , σn Unless, however, the system prescribes for each preference ordering a way to express it, the choice of functions σ1, . . ., σn to characterize honest expression of preferences will be to some degree arbitrary. Hence the choice of a voting scheme to characterize the system will be to some degree arbitrary. Game forms most clearly characterize decision-making systems, but manipulation pertains to voting schemes.... The moral, of course, is that unless a decision-making system prescribes clearly how each voter is honestly to express each possible preferenceordering, manipulation of the system is an unclear notion. A voter manipulates the system if, by misrepresenting his preferences,he secures an outcome he prefers.Unless we have clear standards of honest representation and hence of misrepresentation, manipulation makes no clear sense." Not sure how to work that in, because of course he wasn't talking about LNH specifically, but it is clearly talking about the relationship between strategy and a 1-1 mapping. Homunq (talk) 22:49, 29 September 2012 (UTC)

Also of interest in this context: Endriss, U. “Vote Manipulation in the Presence of Multiple Sincere Ballots.” In Proceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge, 125–134, 2007. http://dl.acm.org/citation.cfm?id=1324268. Homunq (talk) 02:31, 30 September 2012 (UTC)

Hmm.... I should read Sanver, M. R. “Approval as an Intrinsic Part of Preference.” Handbook on Approval Voting (2010): 469–481; I think this may offer some useful cites on this longstanding argument. Homunq (࿓) 11:47, 1 January 2013 (UTC)