# Talk:Ranked voting

## Should cardinal methods be considered ranked methods?

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)

## Conflation of ballot type and tabulation type

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)

## Conceptual overlap of ranked and rated ballots

*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

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 ..."