Should cardinal methods be considered ranked methods?[edit source]
This article is about voting systems that use ranked ballots, which can also include voting systems that use interval scale ballots, i.e. cardinal voting systems
I'd like to see if this is a controversial statement among the Electowiki community. To me, it seems like a bad idea to include rated methods under ranked methods; many people already mistakenly conflate the two categories (i.e. they'll say "rank the candidates from 0 to 5" when explaining Score voting), and this seems to only further add confusion. I think the connection between ranked and rated methods is worth capturing, since a rated ballot is really a ranked ballot with certain constraints and features, but this doesn't seem to be the way to mention that point. BetterVotingAdvocacy (talk) 20:02, 12 April 2020 (UTC)
- I think it needs a wording improvement. What comes to my mind is: "This article is about voting systems that use ranked ballots, although sometimes cardinal voting systems are referred to as using ranked ballots even though they actually use interval scale ballots." Rough wording, but that's the general idea. VoteFair (talk) 01:20, 13 April 2020 (UTC)
- I would go a step futher to distinguish them. I saw the statement above and condidered changing it just yesterday. There are at least three ways I can think of the statement that "Cardinal ballots are a subset of Ordinal ballots" is wrong. In terms of number /group theory they are distinct and do not share overlapping theory. In terms of information theory Cardinal ballots capture more informaiton so at best an argument that "Ordinal ballots are a subset of Cardinal ballots" could be made but I would not think that was useful. In terms of social choice theory it considered different ballot types. To make a statement like this would at least require a source which uses the terms in this way. --Dr. Edmonds (talk) 05:07, 13 April 2020 (UTC)
- I'd actually argue rated and ranked ballots are a subset of a ballot type where you're allowed to indicate your strength of preference in each and every head-to-head matchup between the candidates i.e. for each pair, how strongly you prefer each candidate. So in essence, a rated ballot but allowing you to indicate the scale anew for every matchup. I've written about this at Order theory#Strength of preference to try to define what transitivity might look like with such a ballot. But it seems worth documenting, perhaps on its own separate page, since it is a generalization of choose-one, approval, ranked, and rated ballots, and thus it captures the ideal of the amount of information that a voter should be able to provide in a voting method. Condorcet methods are the only type of voting methods that I can think of that can handle the information offered by such a ballot, though. (To further categorize the voting methods, as I've written before at Score voting#Connection to Condorcet methods, Score and Approval are subsets of Condorcet methods where there is a restriction in place such that the voter's preference can be represented by points, rather than needing to be separated out into individual head-to-head matchups (i.e. if you say A is maximally better than B, then you can't say B is better than C on a rated ballot)). To further elaborate on this, consider that in a matchup between two candidates, putting one at the max score and the other at the min score is equivalent to ranking one above the other i.e. if everyone does this, you just get majority rule. So, with this generalized "cardinal pairwise" ballot, you can indicate ranked preference between every candidate by, for example, "ranking" your 1st choice maximally above every other candidate in matchups, etc. To express rated preference, you just use the same score for a candidate in every matchup they have against another candidate as you would on a rated ballot for them. To express choose-one and Approval preferences, give the candidate(s) you'd mark on those ballots maximal scores in every matchup, and all other candidates minimal scores. BetterVotingAdvocacy (talk) 06:45, 13 April 2020 (UTC)
- BetterVotingAdvocacy I do not really disagree with that. In fact it was my second point, the one about information theory. You could have a ballot system with all pairwise comparisons but instead of just saying the preference you also give the preference strength. This is actually sort of how Distributed Score Voting works theoretically. The important point though is that adding the preference strength is what makes the difference. Cardinal ballots are more generalized (free) and ordinal ballots are more restricted since you cannot give the strength information. The transitivity then becomes additivity and you basically end up with that being the operation of the Group. The interesting thing to realize is that because it is a Group you can combine a set of pairwise strength comparisons into a single score ballot. (Well if we ignore closure but that has only minor implications) However, you cannot combine pairwise ordinal comparisons into a single ranking because of condorset cycles. The ranking does not make a group. Anyway, we could talk about ordinal ballots as being a subclass of cardinal ballots if we really wanted to. I do not see a good motivation to do this as it is more likely to confuse people than to help anybody. In any case this page needs to be cleaned up. Like most electoral stuff on Wikipedia, the heavy hand of FairVote is apparent and we should try to write this with a more neutral description. --Dr. Edmonds (talk) 16:28, 13 April 2020 (UTC)
- It sounds like you're saying that in each pairwise comparison, the voter must maintain the same cardinal preference for a candidate i.e. if my preference is A>B>C and I score, on a scale of 0 to 5, A:5 B:0 in the A vs B matchup, then I can't score B:5 C:0 in the B vs C matchup. What I was talking about was generalized in the sense that you could do that, meaning (as far as I can tell) that the preferences obtained might not be combinable into a rated or a ranked ballot. BetterVotingAdvocacy (talk) 17:55, 13 April 2020 (UTC)
- BetterVotingAdvocacy No, that is what I was getting at with the closure of the group. Lets say the group operation is addition (this way we do not need to deal with infinities like we would with multiplication) so if A:5 B:0 in the A vs B matchup and B:5 C:0 in the B vs C matchup then we need at least a scale of 10 if we wish to put them all on the same score ballot. ie A:10,B:5,C0. I am not really sure what you are driving at here. My point was that the group operation exists in Cardinal systems but does not in ordinal systems so they are very different mathematical objects. This is what makes it impossible for you to give the A:0 C:5 in the A vs C matchup. The cardinal system has some mathematics implicit hiding under it. This is why we can push it all to a single score ballot without loss of generality but pairwise rank and a single ranked ballot cannot really be unified. We do not really need to go down the number theory rabbit hole here. We can on the CES forum if you would like the best books on this are Rudin and Royden but I have not read them in a few decades. The point is that this is not new theory we are talking about. This has been unchanging theory for ages. Ordinal and Cardinal numbers are different. Conflating them is not going to help us in any way. --Dr. Edmonds (talk) 20:43, 13 April 2020 (UTC)
- This is not a very important point, so first off, you are free to skip the discussion on it. But I just want to try to clarify it if possible. So, as an example, let's say a voter maximally prefers A to B. On a rated ballot, it is clear as to how they should express this: put A at the max score and B at the min score. But now let's say they also prefer B to C to some extent. This preference can't be mentioned on a rated ballot, since there is no further room for differentiation when you put the more-preferred candidate at the min score. I am mentioning the idea of cardinal pairwise matchups because it'd allow you to do this, and am further pointing out that a ranked ballot is really equivalent to always putting your more-preferred candidate at the max score and the less-preferred candidate at the min score in each matchup. Thus, this seems a better way to categorize rated and ranked ballots to me than to say that ranked is a subcategory of rated; a ranked ballot doesn't prevent the voter in my example from voting both A>B and B>C while having maximally strong preferences in both matchups. To be clear, this isn't an argument for "ranked ballots are better than rated ballots", but just pointing out that they both capture certain pieces of information that would be lost by converting to the other i.e. a Bernie>Biden>Trump voter with strong preferences between all 3 may not be able to honestly score Biden in between Bernie and Trump without weakening at least one of the matchups, and likewise, ranked ballots can't detect if you only slightly prefer A to B. The generalized cardinal pairwise approach allows one to express both weak preferences in some matchups, and strong preferences in others, so that is why I'm saying it's a useful theoretical concept to help unify rated and ranked ballots conceptually. It is not practical of course to have a voter fill out each and every matchup, but approximations can be done, such as allowing a voter to rate the candidates and then say if they want the weak preferences to be processed, or for each preference to be treated as maximally strong. This is why I mentioned Score being a subset of Condorcet: if you give A 100% support, B 80%, and C 0% on a rated ballot, that is equal to giving 0.2 votes to A>B and 0.8 votes B>C in a Condorcet method. If these preferences are treated as ranked, though, then it is equivalent to giving 1 vote in each matchup to the more-preferred candidate. Sorry for the lengthy response. Edit: It may help to point you to academic articles on this; I don't really understand them well, but I believe the generalized cardinal pairwise preferences I'm speaking about are called "fuzzy pairwise comparisons" in the academic literature. For example, (PDF) https://tarjomefa.com/wp-content/uploads/2015/06/3073-engilish.pdf. Again, I don't understand it all, but I think it gives you a rough idea of what I'm talking about. BetterVotingAdvocacy (talk) 03:30, 14 April 2020 (UTC)
I would strongly advise us to remove this bit. It's confusing and mostly incorrect. Cardinal ballots are fundamentally different from ranked ballots, and interval scales are not the same thing as a ranking. Mathematically, cardinal ballots contain more information than ranked ballots and cannot be treated as subsets: the opposite is mathematically true under the assumption of transitivity. BetterVotingAdvocacy is correct when they state both are subsets of a more general type of ballot which does pairwise comparisons. One can see cardinal ballots as marginals of such pairwise rating ballots, and ordinal ballots as a further discarding of information from cardinal ballots (again, if you assume transitivity). lucasvb (talk} 14:14, 19 July 2020 (UTC)
- I agree that both ranked and Score ballots are subsets of the rated pairwise preference ballot. But further, an argument can be made that a Score ballot is equivalent to multiple fractional ranked ballots, though I don't have an issue with removing the piece you're against. Consider that a Score ballot A:5 B:4 C:3, on a scale of 0 to 5, is equivalent to 0.6 ABC Approval ballots (i.e. a Score ballot of A:3 B:3 C:3), 0.2 AB ballots, and 0.2 A ballots (see KP transform). And obviously, Approval ballots are ranked ballots where the approved candidates are ranked 1st, and the other candidates ranked last. So in some sense, ranked ballots really are a superset of Score ballots, when we make room for fractional ballots. BetterVotingAdvocacy (talk) 16:00, 19 July 2020 (UTC)
Conflation of ballot type and tabulation type[edit source]
I think we conflate many things when we talk about election methods. This community seems to break up the tabulation strategies for electoral methods into two big categories: ordinal and cardinal. We also have two categories of ballots: ranked and rated. The two categories are orthogonal; that is, it's entirely possible to tabulate an election conducted with rated ballots using an ordinal method. In fact, that was my strategy with Electowidget. Moreover STAR voting is a hybrid of ordinal and cardinal tabulation methods. So to answer BVA's question: I believe that it would be difficult to tabulate ranked ballots using cardinal methods, though I suppose that's what the Borda count is. -- RobLa (talk) 05:16, 13 April 2020 (UTC)
- Over on the talk page for the "Ranked ballot" redirect (at Talk:Ranked ballot), User:BetterVotingAdvocacy suggested changing the redirect from this state:
- ...to this state:
- That might be acceptable, but my fear is that the conflation problem I ^described on April 13 is being ignored. Moreover, I'll note that in the current wording of Ballot#Ranked ballot, the first words are:
See Preferential voting.
- Why doesn't that say this?
See Ranked voting.
- For that matter, why do we have separate articles for Ranked voting and Preferential voting? It seems we need to simplify our collection a little bit before we get too excited about expanding it. Moreover, we need to agree on a taxonomy: am I wrong for suggesting that this is the correct taxonomy?
- Ballot type
- Ranked ballot
- Rated ballot
- Tabulation type
- Ordinal method
- Cardinal method
- As I stated in my April 13 comment, it's possible to have rated methods tabulated using an ordinal method, and ranked methods tabulated using a cardinal method. I haven't gotten either acknowledgement that my proposal is more-or-less correct, nor objection to my proposed taxonomy, so I'm not eager for us to make changes that ignore my proposal. -- RobLa (talk) 20:36, 20 April 2020 (UTC)
- I had assumed you created the Ranked voting article knowing that there was already a Preferential voting article. [Next point] I don't think that there really is a strict "ranked ballot/ordinal method" dichotomy for naming; I refer to cardinal methods all the time as "rated methods" and have seen people call rated ballots "cardinal ballots", so I'd like to see the same categories and taxonomy you mentioned, but let the words rated/cardinal and ranked/ordinal be interchangeable. BetterVotingAdvocacy (talk) 22:23, 20 April 2020 (UTC)
- I created the "Ranked voting" article because I saw that Wikipedia has two articles:
- I had admittedly forgotten that I had copied over the intro to "Ranked voting" over from Wikipedia, but now I remember my rationale. Our "Preferential voting" article has drifted too far away from its Wikipedia counterpart that we run the risk of confusing our readers. We need to figure out how to align our content with Wikipedia's content. The Preferential voting article probably needs to get merged into the Ranked voting article, and "Ranked voting" probably needs to be the main article. Looking at the edit history for Preferential voting on Electowiki, it appears as though it was copied over from a 2005 version of the Wikipedia article. The edit history for wikipedia:Ranked voting is harder to read, but it looks like there has been a concerted effort to define an equivalence between "ranked ballot" and "ranked choice voting" over the past few years . Given that "Preferential voting" is still the term used to describe Instant-runoff voting in some parts of the world, we have a confusing soup of terminology.
- I believe that we need a single article to describe the ranked ballot independent of tabulation method. Whether we call that thing "Ranked" or "Preferential" (or even "Ordinal") is less consequential to me than that we have something that we use consistently, and that we use it to describe the ballot, not the tabulation method. Since Wikipedia seems to have settled on "ranked voting" as the term to generically describe voting using ranked ballots (and currently redirects wikipedia:Ranked ballot to the "Ranked voting" article), I'd prefer to call it "Ranked voting". But regardless, let's make sure that when we differ from Wikipedia, we're deliberate about it. -- RobLa (talk) 23:34, 20 April 2020 (UTC)
Conceptual overlap of ranked and rated ballots[edit source]
I think the connection between ranked and rated methods is worth capturing, since a rated ballot is really a ranked ballot with certain constraints and features, but this doesn't seem to be the way to mention that point. -- (quote from "#Should cardinal methods be considered ranked methods?" by User:VoteFair at 01:20, 13 April 2020 UTC)
- Agreed. It could be mentioned that there is conceptual overlap, but saying that one is a sub-type of the other is not really correct or instructive. — Psephomancy (talk) 02:45, 14 April 2020 (UTC)
False statement about strength of preference[edit source]
The following recently-added statement is not true. It does not apply to the Condorcet-Kemeny method, nor to the Instant Pairwise Elimination method.
"... if a voter ranks X>Y>Z, then the strength of their preference for X>Z must be stronger than their preference for X>Y or Y>Z, yet all 3 preferences are generally treated as equally strong in most ranked methods ..."
- The final part of the statement says "most ranked methods", not all. BetterVotingAdvocacy (talk) 22:16, 14 April 2020 (UTC)
- I agree with User:VoteFair that this sentence is misleading, and moreover, the new "Notes" section is problematic. I'm tempted to remove the entire "Notes" section, because it lacks citation (of either a peer reviewed paper, or an open online discussion). Maybe what is written there makes sense to someone with an advanced math degree, but it doesn't make sense to me. Particularly the part where it says "In some sense, this can be seen in the ranked context by first using the KP transform..." is troublesome. User:BetterVotingAdvocacy, could you more deliberately avoid accidentally engaging in proof by intimidation? I think if we're going to introduce readers to the Kotze-Pereira transformation, we need to provide a lot more context that just a link (and honestly, I'm not sure that linking to KP transform helps clarify anything). -- RobLa (talk) 22:28, 14 April 2020 (UTC)
- I will take a second stab at fleshing out the section and providing background for the KP transform, and if it still isn't good enough, feel free to remove the section. Edit: I'm not sure it really is possible to give background for the KP transform on second look; I'd ask you to, if you're considering removing the whole section, think about removing the paragraph discussing the KP transform while leaving the rest of the Notes section intact, if that is okay with you. BetterVotingAdvocacy (talk) 22:40, 14 April 2020 (UTC)
- Thanks, User:BetterVotingAdvocacy. In revision number ("oldid") 10376, I changed the prior "Notes" section (there were two) into a subsection titled "Strength of preference". I nested that in a new section titled "Criticisms". That sounds like a big change, but as you can see from the difference listing ("diff") of oldid=10376 with the previous revision, this was a pretty small change to the text of the article. There's a lot more restructuring that we need to do to the article, since the introduction section (before the table of contents) is way too long. The Wikipedia counterpart that I copied it from needs similar work. The new "Strength of preference" section should probably be a lot shorter, too. -- RobLa (talk) 18:36, 15 April 2020 (UTC)
Majority rule as an approximation of utilitarianism[edit source]
User:Lucasvb, I have some issues with your recent edit. I do agree with the conclusion but I am not sure that it follows from the logic given. Do you have a reference for this? Is this an accepted belief in the academic field?
My main qualm is that I do not see how the example of a dictator really fits into the argument. Maybe there is nothing the people have to offer the dictator but how is this related to the relation between majoritarianism and utilitarianism. Utilitarianism is not the same as the free market because utilitarianism assumes the equality of each persons total utility.
You state This is what majority rule is doing. It is used to justify the violation of preferences of a minority (like the sole dictator) in order to pursue a "better" equilibrium (the majority of the population). but this is also what Utilitarianism would do. The difference is that Utilitarianism takes into account the strength of preference. Majoritarianism has the preference strength the same for all people so it is a first order approximation of a Utilitarianism.
The closest argument to yours I have ever been able to find is from Nobel laureate Milton Friedman in this lecture. He gets to the point pretty quickly so you only need to watch the first 15min or so. In it he says that Majority rule (Majoritarianism) is an expedient to Unanimity. He alludes to the free market as a utilitarian mechanism because of the Pareto exchange mechanism. Of course I agree that this only works when all people have the same skills, ability and prior wealth but he is correct because of the Pareto equilibrium. So he does not draw the line that you do. --Dr. Edmonds (talk) 16:28, 8 July 2020 (UTC)
- His arguments can be found in more length at https://www.reddit.com/r/EndFPTP/comments/hi29r8/discussion_of_a_vote_form_an_score_example/fweu53r/ and https://www.reddit.com/r/EndFPTP/comments/hi29r8/discussion_of_a_vote_form_an_score_example/fwey9w4/. BetterVotingAdvocacy (talk) 18:23, 8 July 2020 (UTC)
- In the second comment, he says "So, you are assuming "1 unit of potential violence" for every person equally, assuming you can aggregate those units, then assuming the side with "more violent potential" wins.
- You are assuming the power to be violent and the amount of violence inherent in each person is equal and commensurable, and aggregating it."
- I think that you misunderstood things. The argument here is about ordinal vs cardinal utilities, not majoritarianism vs utilitarianism, and I'm certainly NOT claiming the Pareto equilibrium is utilitarian (which is incorrect, as the dicator example clearly demonstrates), only pointing out that the Paretian ideas are present in the economics literature.
- The point is that under strictly ordinal preferences (if you are working ONLY within an ordinal utility framework, like economists do), arbitrarily anti-democratic situations must be considered acceptable, as Pareto equilibrium (the best you can do under such framework) implies you cannot violate the preferences of even a single individual, like the dictator, when changing social states. Every change has to be unanimously positive or neutral.
- This shouldn't be controversial, as it's all pretty standard in the ordinal vs cardinal utility literature.
- This argument is usually presented inverted in the literature (typically right-wing Libertarian), when people use Paretian principles to criticize democracy as a form of social organization and promote free market principles as an alternative: "democracy is two wolves and a sheep deciding what's for dinner", they say. To prevent the "mob rule" from violating even a single individual's rights, one invokes Pareto equilibrium: the wolves cannot get what they want without violating the preferences of the one sheep, so if you claim to defend individual rights absolutely you need every single individual to either agree or be indifferent to a change in the social state. Since the sheep doesn't want to get eaten, society just lives with the status quo and the sheep lives. This is seen as a triumph of "free-market" (but in actuality Paretian) principles over "mob rule" (democracy).
- This is precisely the same argument I've outlined, but now the sheep (dictator) is the villain of the story. If you wish to change to a different social state, you necessarily must violate the preferences the dictator has to remain in power. You also need an additional justification outside your framework to do so. So a Paretian framework necessarily invalidates any move towards the will of the majority, or any sense of democracy. To navigate the Pareto frontier towards a "better outcome" one must invoke some additional principles.
- The only solution out of this is to invoke some notion of CARDINAL utilitarianism, by claiming that every individual has equal claim to violate anyone else's will (I referred to "violence" in those reddit posts as that was the argument being used), and so the largest group thus decides which minority preferences to violate in their favor. This is majority rule, but it is fundamentally a cardinal utilitarian argument as you necessarily must aggregate multiple individuals preferences in a cardinal way. Under strictly ordinal preferences, there is no such concept as "counting"!
- Another point that seems wasn't clear is that cardinal utilities doesn't mean "non-uniform strength of preferences". If everyone is assumed to have exactly the same utility and you add them up, you're still operating under a cardinal utilitarian framework, you're still "adding multiple individuals".
- The claim that my A>B cancels your B>A ("every individual has equal claim to violate anyone else's will") is a cardinal claim, and this is underlying majority rule. You are simply assuming every ordinal preference has exactly the same cardinal utility difference when comparing them. Perhaps this can be edited to make the argument cleare, and I could fetch some references for specific points later on. lucasvb (talk} 17:43, 18 July 2020 (UTC)
- User:Lucasvb This is the first I am hearing of ordinal utilities. I do not see how this term is useful. Utilities are generally thought of as a numeric measure. Ordinal systems only provide a preference direction not a magnitude. this term conflates preference with Utility. Is this term used in literature anywhere? I think you just mean preference when you say ordinal utility. If that is the case then why not use the less confusing term? --Dr. Edmonds (talk) 21:22, 18 July 2020 (UTC)
- That's surprising, as it's THE standard term used everywhere, and it's even already mentioned in the article's reference section. https://en.wikipedia.org/wiki/Ordinal_utility or https://scholar.google.com/scholar?hl=en&as_sdt=0,5&q=ordinal+utility ... Utilities are what individuals use to make decisions. I don't like the term either, but since it exists we cannot use "utility" without clarification. lucasvb (talk} 09:33, 19 July 2020 (UTC)
- I disagree. To me, the claim that ordinal voting is really cardinal feels like the (obvious) observation that any ethics system can be cast as a form of utilitarianism, because every ethics system cares about some values and disregards others. For instance, you can make deontology utilitarian by saying that the adherents assign utility -infinity to going against their duty; or you can make Rawlsian justice utilitarian by considering it a variant of minmax utilitarianism. However, neither deontology nor Rawlsian justice are, to my knowledge, usually considered to be utilitarian.
- So what's the logic of (majoritarian) ordinal voting? Well, it's these two things:
- 1. Each person's voice is of equal value (this can be justified by some kind of veil of ignorance or birth lottery argument without importing the whole machinery of total utilitarianism).
- 2. Beyond that, we don't know what the voters' relative scales are (e.g. if A>B does that mean A: 1000, B: 1, or A: 10, B: 1?), and we have no basis for interpersonal comparison (is my first preference stronger than yours?).
- There are other variants of ordinal voting that are not majoritarian, and thus discards the first point here. For instance, the Heitzig consensus scheme I've been talking about recently on the EM list is closer to the Rawlsian approach, although it isn't quite the same thing. While the scheme is not majoritarian, I'd still consider the voting done in it ordinal, however. Kristomun (talk) 09:45, 21 July 2020 (UTC)
- Thinking about this a bit more, I suppose all ordinal methods are, in the sense that say Plurality or Borda is, cardinal, because their algorithms use numeric variables. But by that criterion, every deterministic, neutral, anonymous, and resolvable voting method is "cardinal". What the argument above shows is that ordinal methods are not necessarily utilitarian, or approximately utilitarian, unless every system of ethics that can be cast in a utilitarian form is also utilitarian. Kristomun (talk) 11:18, 21 July 2020 (UTC)
- Kristomun That is well put. I think electowiki needs a page on this topic. A sort of comparison between the majoritarian and utilitarian philosophies which underpin different systems. I tried to give an explanation of a specific point of that here. Are you interested in giving it a shot. I have always been hesitant because I do not know where to start.
- On a different topic are there any Rawlsian minmax electoral systems? I tried to use the total number of unspent points as a metric in PR systems to measure quality. ie MAX-sum(score) for each voter is the amount of unspent score. So you want to try to minimize the total amount of unspent score. This is basically what Sequentially Spent Score does with its reweighting but the selection is pure Utilitarian. --Dr. Edmonds (talk) 15:42, 21 July 2020 (UTC)
- There are the so-called "consensus" or "minmax" multiwinner methods, like Minimax approval. Let a voter's satisfaction with a council be his max score of a candidates on that council. Then minmax Range chooses the council so as to maximize the minimal satisfaction with that council. Approval is easier, it just minimizes the maximal Hamming distance. Since min and max are not robust statistics, these methods are vulnerable to strategy.
- In addition, I'd say unanimity-based voting would fit, because if the worst-off voter doesn't agree to the proposal, it doesn't happen. However, high supermajority and unanimity voting have a status-quo bias, which is what the Heitzig mechanism I've referred to tries to do away with.