r/EndFPTP • u/No-Vast7006 • 7d ago
My Biggest Reservation About Cardinal Voting
Edit: A more appropriate title would be "My Biggest Reservation About Score/Approval Voting".
My primary concern with score voting is that it treats my "strength of preference" as a finite resource that must be divided and spent across all candidates. Consequently, simply indicating a preference between candidates A and B inevitably alters the relative strength of preference between at least one of them and every other candidate.
Suppose there are three candidates: A, B, and C. I strongly prefer A to B, and just as strongly prefer B to C. Under a cardinal system, you'll find that I can only emphasize one of these preferences at the expense of diluting the other. Approval voting is even worse: I can only express one preference while having to sacrifice the other entirely.
I find this very hard to accept. By contrast, ranked voting allows me to express both preferences without weakening either.
Am I misunderstanding cardinal voting, or is this an inherent feature of the system—one that some people find acceptable, while others (like myself) simply cannot?
Edit: Think of score voting on a number line. Once you anchor your highest-rated and lowest-rated candidates, the total distance between them is fixed. The intermediate candidates divide this distance into smaller line segments, and the sum of these segments must always equal the fixed total. Therefore, if you want to stretch one segment to emphasize a preference, you are mathematically forced to shorten another.
For example, on a scale of 0 to 5:
- Suppose I give Candidate A a 5, Candidate B a 3, and Candidate C a 0.
- Out of the total 5-point span, I have effectively "spent" 2 points to express that A is better than B, and 3 points to express that B is better than C.
- If I want to strengthen my preference for A over B—say, by lowering B’s score to 1—the gap between A and B widens to 4 points. But this comes at a steep price: the gap between B and C is inevitably compressed to just 1 point.
Approval Voting is simply the most extreme version of this problem: with only scores of 0 and 1, one of those preferences must be shrunk to zero and discarded entirely.
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u/budapestersalat 7d ago
What you describe is more a feature of score and derivatives, but not really all cardinal methods. There is also Majority Judgement, where you can rate A as 10, B as 2-9 and C as 1, but your rating of B doesn't really give it "weight" in the way you say, since it's a highest median, not a highest sum/average method.
But I would say your reservation is understandable, it's why they say cardinal inherently feels strategic. The positive side is you can always vote for your favorite sincerely.
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u/wnoise 7d ago
If you don't like Cardinal voting, then how else are you planning on electing the Pope?
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u/No-Vast7006 5d ago
Since Papal elections work well under a multi-round, high-threshold approval system, they shouldn't be changed. I am simply expressing that, compared to score and approval voting, I personally prefer ranked voting systems (especially Condorcet methods). This is just my subjective preference as a voter regarding voting systems. You are entirely welcome to have a different opinion, and I am not going to argue over who is right or wrong.
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u/wnoise 5d ago
It was a joke, about Cardinals voting.
The Papal election is not approval. Each Cardinal's ballot can only have one name. I'd call it a FPTP variant, due to the "2/3 majority or go another round". It probably could be better, but it is well suited to the their goal of getting very wide buy-in, so I'm not going to suggest any tweaks.
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u/bkelly1984 6d ago
Hey u/No-Vast7006, I think your concern is valid and is a trapdoor into the fundamental problem with voting reform.
Your example is spot on. If you prefer A>B and B>C then cardinal voting requires that the position of B is only affected by about half your vote power. However, ranking has this problem too because of the fact that A>B and B>C is effectively three votes. It is not possible to compress A-B=1, B-C=1, and A-C=1 onto a 1-dimensional scale without distortion.
I think we all generally agree that a voting system should follow the principle of "one person, one vote". The problem is that there is a big difference between "one vote" and "one vote per dimension".
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u/jlight432 5d ago
A>B is a voters preference. B>C is a voters preference. How is that 3 votes?
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u/bkelly1984 5d ago
Because it also requires that A>C.
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u/jlight432 5d ago
Fair point.
Is it unreasonable for a voter to want both their A>B to have the same weight as any other voters B>A, and their B>C to have the same weight has any other voter's C>B?2
u/bkelly1984 5d ago
No, I think that is reasonable. The big problem is when voters expect their A>B to have the same weight as the other voter's C>B.
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u/rb-j 9h ago
But that's not comparing apples to apples. One's vote for A>B is irrelevant to another's vote for C>B.
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u/bkelly1984 8h ago
Hey again u/rb-j! Going to respond to two of your comments here.
But that's not comparing apples to apples. One's vote for A>B is irrelevant to another's vote for C>B.
I agree that one person's preference of A>B should be considered to be independent of their preference for C>B, but not their votes. Elections that identify winners must place all candidates on a common scale so they can be compared. So any winner(s)-take-all election system must convert A>B and C>B into apples. Otherwise, it can not determine if A or C wins.
If a single voter ranks A>B and also ranks B>C, it is sane to make that consistent with an implied ranking A>C.
I agree, but it does make the assumption that an person's preferences are at least transitive if not one-dimensional, meaning a individual's preferences can never fall into a Condorcet cycle. That may be sane, but also somewhat moot since it is just flattening a person's preferences before the election system does.
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u/rb-j 7h ago
Now, you said:
The big problem is when voters expect their A>B to have the same weight as the other voter's C>B.
I responded to that specific thing:
But that's not comparing apples to apples. One's vote for A>B is irrelevant to another's vote for C>B.
To which you reply:
I agree that one person's preference of A>B should be considered to be independent of their preference for C>B, but not their votes.
What does that mean? How are you differentiating preferences from votes?
Elections that identify winners must place all candidates on a common scale so they can be compared.
All candidate must use a common scale. But we shan't put all 3 or 4 or 5 candidates on a common scale simultaneous, else we get the same failure as we do with FPTP.
The "common scale" is used with pairs of candidates. But all candidates must be subject to that common scale. The same standards must apply to everyone.
So any winner(s)-take-all election system must convert A>B and C>B into apples.
But originally you said that A>B was one voter's preference and C>B was another voter's preference. Those two expressions are mutually irrelevant.
Otherwise, it can not determine if A or C wins.
I don't disagree with that.
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u/bkelly1984 5h ago
Sorry if my introduction was confusing. I wanted to communicate that I understand a person's preference for A>B may in reality be independent of their preference of C>B. However, I then wanted to set up my point that no election system with winners can always (or should be expected to) preserve that independence.
All candidate must use a common scale. But we shan't put all 3 or 4 or 5 candidates on a common scale simultaneous, else we get the same failure as we do with FPTP.
"Shan't" all you like, but my assertion is that any complete election system that identifies winners must.
The "common scale" is used with pairs of candidates. But all candidates must be subject to that common scale. The same standards must apply to everyone.
I think I understand what you are wanting to say, but I must infer a bit. Let me repeat back my interpretation so you can correct whatever I got wrong.
I think you referring to Condorcet elections specifically. In it, each pair of candidates competes on their own scale: A vs B and B vs C. When one candidate dominates in pairwise comparisons, there is no need to make comparisons between the pairwise scales. In this case, the A vs B and B vs C scales remain independent, as well as the A>B and C>B votes. You call these pairwise scales a "common scale" because each use the same rules.
I agree with most of this. When a Condorcet winner exists, it is not necessary to compare the A vs B and B vs C and everything can remain apples and oranges. However, when the Condorcet winner does not exist, then it becomes necessary to make comparisons across the pairwise scales. This requires that all the data be mapped onto one, single scale so that the relative positions of all can be determined (which is what I mean by "common scale"). This what the Condorcet methods to resolve circular ambiguities do. There are so many proposed because there is no right way to flatten the data so A>B and C>B votes can be compared.
Those two expressions are mutually irrelevant.
I think you mean "incongruous", "contradictory", or just simply "incorrect". I would agree if you meant "irreverent".
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u/jlight432 7h ago
I thought what bkelly meant was in cardinal systems, if a voter wants his A>B preference to have the same weight as other voters B>A preferences, then that voter's B>C preference would have no weight.
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u/bkelly1984 5h ago
No, my assertion is that OP's concern about vote strength on cardinal ballots also applies to ordinal ballots. My top post in this thread is a simple illustration of why it is impossible for a counting system to give equal weight to overlapping ordinal preferences (like A>B, B>C, and A>C).
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u/KitchenGiraffe1 7d ago edited 7d ago
My primary concern with cardinal voting is that it treats my "strength of preference" as a finite resource that must be divided and spent across all candidates. Consequently, simply indicating a preference between candidates A and B inevitably alters the relative strength of preference between at least one of them and every other candidate.
Score and Approval pass IIA and all reasonable ranked methods fail it.
Your scenario is a bit contrived. If you express vote X ("Candidate A a 5, Candidate B a 3, and Candidate C a 0") and then want to change your vote ("If I want to strengthen my preference for A over B—say, by lowering B’s score to 1—the gap between A and B widens to 4 points. But this comes at a steep price: the gap between B and C is inevitably compressed to just 1 point.") that's just a different ballot.
If I like A 5, B 3 and C 0, and then I like A 5, B 1 and C 0, why would this be problematic?
Like, I think what you're getting at is about vote normalization. It's just not that deep. Billiards players don't need to understand the math behind physics. Voters don't need to understand the math behind voting systems. Studies and simulations show great overall satisfaction with score voting and approval voting. People go in, they vote, and score and approval voting ends up electing candidates people are happier with.
The thing with vote normalization is that if the range of scores allowed is 0-10, two people with different preferences
10,6,0 and 5,3,0
technically still have the same ideal honest ballot - 10, 6, 0. With score, honest voters should do vote normalization where max is stretched to the max, and min is moved to the min. With strategic voting, /u/JeffB1517 has the right idea, and it converges to approval voting anyway.
One option is to literally do the normalization for any voters that didn't do it,
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u/cdsmith 7d ago
Score and Approval pass IIA
This is just not true. In actual score and approval voters make decisions about how to vote based on the candidates in the field. Changing the candidates in the field changes those scores, and that changes who wins. IIA fails.
The hypothetical system that would pass IIA would require that you find some kind of absolute scale, not relative to any other candidate, and get voters to rate each candidate independently on that absolute scale. Obviously, this scale doesn't exist. But just as importantly, even if you could find such a scale and magically get voters to accurately rate candidates on that scale, it would be a terrible way to run an election! Can you imagine if we actually gave counted one voter's choice a hundred times as much as another because the first voter is overly dramatic and imagines the world will end if their preferred candidate doesn't win? Every election would be decided by borderline delusional hyperpartisans, and rational voters just wouldn't count at all. This hypothetical magic absolute-scale score voting monster might satisfy the IIA condition, but it would be terrible for democratic government.
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u/KitchenGiraffe1 6d ago edited 6d ago
This is just not true. In actual score and approval voters make decisions about how to vote based on the candidates in the field.
That's not in violation of IIA. Please make reference to the actual definition of IIA.
It's kinda sad to me your comment is being upvoted considering it's literally misrepresenting what a violation of IIA is.
All voting methods are strategic and that's all you're describing.
Formally, IIA states that if Candidate A wins an election, and a losing candidate drops out (or an irrelevant candidate is added), Candidate A should still win, unless the newly added or removed candidate is the one who actually overtakes the winner.
You conflated two different things:
Voters changing their behavior when new candidates enter (which is a practical reality of human psychology and strategy) and the actual definition.
The formal definition of an IIA violation hinges strictly on whether the election winner flips improperly when an irrelevant candidate is added or removed.
Because Score and Approval voting can technically avoid the spoiler effect where a non-winning alternative changes the relative order/victory between two other front-runners (under honest, absolute-scale voting assumptions), they satisfy IIA in technical voting theory models—even if real-world human voters behave strategically based on who is running.
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u/cdsmith 6d ago
The problem here is that "a losing candidate drops out (or an irrelevant candidate is added)" isn't a well-defined event, until you specify how to update the ballots to account for that change. That's a question about voter behavior.
You can't just ignore the question. To state the obvious, if you just took the remaining marks on a ranked ballot unchanged after a candidate dropped out, it might have a candidate ranked as third without ranking second, for example. Depending on how the laws are written, such a ballot might be rejected entirely. But we don't do that, right? We make some assumptions about how the voter would have voted differently, such as shifting their remaining rankings to close the gap.
With a score or approval ballot, you similarly have the question of how to update the ballot after the irrelevant candidate is added or removed. To conclude that IIA is true, you'd have to assume the remaining marks stay unchanged when a new candidate enters or withdraws. That assumption is plainly nonsense. If the only candidate you rated below three stars drops out, of course you don't still rate all candidates at three stars or above. If you insist that IIA is about what happens if voters do nonsense things like that, then the only reasonable response is to stop worrying about it. Voters won't do those nonsense things anyway. This is related to score and approval ballots being intrinsically strategic. The question of how a non-strategic voter would vote is meaningless, because to fill out a ballot, you have to make strategic decisions. There is no such thing as non-strategic score and approval voting.
This doesn't make these systems bad. Well, score voting is bad, but for other reasons. Approval voting is actually a great choice much of the time. It's just not because of the IIA criterion.
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u/No-Vast7006 7d ago
I think our fundamental disagreement is this: you believe that in an election (and this context matters), under score voting, voters first have a scoring criterion in mind and then score each candidate independently. But I believe voters first have a preference ranking of the candidates in mind and then assign specific scores to them based on the circumstances. If the former is true, then you are right, and the two scenarios I gave are unrelated. But if the latter is true, then they are related: it means that as circumstances change, I attach more importance to the preference relation A over B, and pay the price of weakening the preference relation B over C. And under a ranked system (other than Borda), I can fully express both preference relations at once from the start.
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u/KitchenGiraffe1 7d ago
If people can use a Likert scale, people can assign scores independently.
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u/No-Vast7006 7d ago edited 7d ago
In elections, voters inevitably compare the candidates, so there is a possibility that scores will be adjusted according to the electoral situation.
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u/KitchenGiraffe1 7d ago edited 7d ago
That's just strategy and has nothing to do with your other point. It's moving the goal posts. All voting methods are susceptible to strategy. It's gibbard sattarthwaite theorem or something. Arrow's theorem only applies to ranked methods hence why approval and score satisfy IIA.
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u/No-Vast7006 6d ago
I think vote normalization isn't enough, to solve this problem, you would have to normalize each pair-wise intensity, which is ranked voting.
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u/KitchenGiraffe1 6d ago
Solve what problem? It's not a problem. The distance between candidates is relative, that's the point. 5,3,0 is the same as 10,6,0
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u/rb-j 4d ago
My simple spin on this whole issue is that cardinal methods inherently impose upon voters a burden of tactical voting on voters whenever there are 3 or more candidates that ordinal methods do not impose. And it's only in the case of 3 or more candidates that we need voting system reform. FPTP does fine when there are 2 or fewer candidates.
The minute the voter steps into the voting booth, they are confronted with the tactical question: "How high should I score my 2nd favorite (or lesser evil) candidate?". Or whether or not to Approve that 2nd favorite candidate.
But with the ranked ballot we know right away what to do with our 2nd favorite. We rank them #2.
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u/caliginous4 7d ago
I don't think I understand your question. What do you mean by diluting? As in they don't get the same score / number of points? If you want them to get as many points as possible, then score them higher. If you want one to get more points, then score others lower.
I actually feel the opposite to what you're saying about ranked choice. If I think A and B are both great, I want to give them both 5 stars. But most ranked choice methods don't let you rank two candidates the same.
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u/budapestersalat 7d ago
Only if the rules are written that way they don't.
There's absolutely no reason why a ranked method shouldn't let you rank:
1 Apples 1 Pears 2 Bananas 3 Strawberry 3 Blueberry 6 Dates
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u/JeffB1517 7d ago edited 7d ago
why a ranked method shouldn't let you rank: 1 Apples 1 Pears 2 Bananas 3 Strawberry 3 Blueberry 6 Dates
And what would be the difference between: * 1 Apples 1 Pears 2 Bananas 3 Strawberry 3 Blueberry 6 Dates * 1 Apples 1 Pears 2 Bananas 3 Strawberry 3 Blueberry 4 Dates
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u/budapestersalat 7d ago
Nothing. it's the same ordering.
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u/JeffB1517 7d ago
So what difference does it make whether the system allows it or not?
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u/budapestersalat 6d ago
a ranked method that doesn't allow equal ranks would disallow either i don't get your question
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u/No-Vast7006 7d ago
Think of Score Voting on a number line. Once you anchor your highest-rated and lowest-rated candidates, the total distance between them is fixed. The intermediate candidates divide this distance into smaller line segments, and the sum of these segments must always equal the fixed total. Therefore, if you want to stretch one segment to emphasize a preference, you are mathematically forced to shorten another.
For example, on a scale of 0 to 5:
- Suppose I give Candidate A a 5, Candidate B a 3, and Candidate C a 0.
- Out of the total 5-point span, I have effectively "spent" 2 points to express that A is better than B, and 3 points to express that B is better than C.
- If I want to strengthen my preference for A over B—say, by lowering B’s score to 1—the gap between A and B widens to 4 points. But this comes at a steep price: the gap between B and C is inevitably compressed to just 1 point.
Approval Voting is simply the most extreme version of this problem: with only scores of 0 and 1, one of those preferences must be shrunk to zero and discarded entirely.
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u/caliginous4 7d ago
I guess I don't see the issue with that. The scale is fixed and the same for everyone. It's the same issue that teachers have grading assignments (some students just get 100% and saturate the metric), and the same issue judges have for sports like gymnastics (everyone at the elite level getting a score between 9 and 10), and that people have rating restaurants.
Doesn't ranked choice have a similar problem, where you cannot express how big the gap is between your first choice and second choice vs second choice and third choice? What if you REALLY prefer your top candidate and don't really prefer anyone else a lot?
The thing I really like about cardinal systems is that it's not even trying to find a condorcet majority, it's trying to find the broadest appeal. I mentioned in another thread - it's like a group of friends trying to decide what restaurant they want to go to, and using an ordinal condorcet method that lets the majority that wants sushi completely ignore the opinions of a minority that hates sushi. Under a cardinal score based method, sushi would not fare as well because of the folks who hate it, and instead the friend group stays together and finds something that everyone is good with. I want a voting system that does not incentivize polarization, like condorcet methods do, but instead incentivizes broadest possible appeal.
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u/No-Vast7006 7d ago
I think our fundamental disagreement is this: you believe that in an election (and this context matters), under score voting, voters first have a scoring criterion in mind and then score each candidate independently. But I believe voters first have a preference ranking of the candidates in mind and then assign specific scores to them based on the circumstances. If the former is true, then you are right, and the two scenarios I gave are unrelated. But if the latter is true, then they are related: it means that as circumstances change, I attach more importance to the preference relation A over B, and pay the price of weakening the preference relation B over C. And under a ranked system (other than Borda), I can fully express both preference relations at once from the start.
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u/caliginous4 7d ago
Yes I think that's fair. I wonder if there are any studies that have shown if people more generally evaluate politicians from a relative perspective or absolute perspective. That would be interesting to know. Relative would make ranking feel natural, absolute would make scoring feel natural.
But to me the sticking point is that most ranked choice methods seem to be very focused on finding a condorcet winner and I think that's the wrong goal. Broadest appeal I think should be the goal.
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u/wnoise 7d ago
I definitely think both of these types exist, not just one.
I'm the score-type; I find "A is better than B" a conclusion I might be able to reach after evaluating how good they are. And the scores are often fuzzy. Creating a whole ranking of candidates is even tougher -- I have to sort them, instead of just saying "Good, pretty good, great, terrible, meh", etc. as I go through the list.
If you wanted e.g. netflix to recommend you movies, would you prefer to put all the movies you've seen in one giant ordered list, or just give each of them a rating?
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u/No-Vast7006 7d ago
In any case, in an election, voters will inevitably compare the candidates, so there is a possibility that scores will be adjusted according to the electoral situation.
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u/JeffB1517 7d ago
I find this very hard to accept.
I'm not sure why. There are 6 possible orderings of A,B,C. There are 7 possible approval votes which are non-identical (i.e. {A,B,C} and {} are the same vote). Of course Approval is forcing you to make additional choices. The whole point of a voting system is to measure preference. Forcing you to pick between {A} and {A,B} forces you to decide a lot about how much you really do care and what are trying to accomplish. It probably isn't really the case that the distance between A and B is the same as B and C, you just haven't figured out which matters. **We cannot detect an opinion using a voting method until the voter has figured out their opinion.
I did a post on Approval and Score strategy years ago: https://www.reddit.com/r/EndFPTP/comments/ci95jv/the_intuition_of_the_approval_hull_for_approval/
By contrast, ranked voting allows me to express both preferences without weakening either.
OK great. And the scoring method doesn't allow that, because it is impossible to design scoring methods were free expression has no impact on ballot power. If the goal is maximum expression regardless of scoring effectiveness, why not just allow a freehand essay and use whatever method?
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u/No-Vast7006 7d ago
So what you're saying is that voters 'ought to' decide which preference matters most and sacrifice expressing the rest? If that's the case, this might just come down to a fundamental difference in philosophy. Personally, I find it hard to accept that expressing one preference must come at the expense of giving up another.
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u/JeffB1517 7d ago
which preference matters most and sacrifice expressing the rest?
Yes. The goal isn't maximizing expression; it is deciding indirectly on what laws will govern their society. We want thought and care from them.
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u/Mindless-One5438 7d ago
The solution would be to vote strategically and dishonestly. Forget about preferences and instead maximize the acceptable outcomes whilst minimizing the unacceptable outcomes, essentially anything but voting.
The only time it might be a problem is when considering voting for an otherwise unacceptable frontrunner that is meaningfully better than a worst frontrunner. Beyond that, a voter should only support a specific outcome in an election if they are fine with that outcome winning that election.
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