Cardinal voting is a lot more expressive than ordinal. For example, with 3 candidates, the following 3 cardinal votes

A:35% B:33% C:32%

A:99% B: 1% C:0%

A:51% B:49% C:0%

all reduce to the same ranking but express rather different relative preferences. Btw, in this example scores are normalized to add up to 100%, but should still be considered independent of other candidates.

So what we should actually discuss more is how preferences are embedded. The problem with ordinal systems are that exactly the same but with cardinal systems they aren't. I don't think a percentage system is a good way for expressing the advantage because people are going to be used to more discrete systems like a 5 star rating system. Here's a better example:

Suppose there are three candidates, A, B, C, and D. We have a strong preference for A, we like B and C equally, and strongly dislike D. How do we express these?

- Ordinal: A > B > C > D

- Cardinal (5 points): A:5, B:3, C:3, D: 0

We can see the embedding issue here pretty clearly. In an ordinal system there cannot be a means for us to express that we have no strong preference of B over C. We also can't express how much we like C over D because we're forced to place them equidistant from one another. This is a big problem because in the ordinal system we say that our preference of B over C is identical to our preference of C over D. There is a clear advantage here to cardinal systems that have a lot of downstream effects that lead to cardinal systems passing the favorite betrayer criteria -- aka strong spoiler -- (ordinal fails) and is why these systems are monotonic (ordinal systems aren't).

I argue strongly in favor of cardinal systems but I often find that people are strongly invested in ordinal systems. Unfortunately fair vote has done a great job at advocating for RCV and people get passionate about things they are not passionate about learning about. But I need to note that there are downsides to cardinal systems too. Voting systems are over constrained and so it really depends on what criteria you are optimizing for. If we strictly want the best VSE over everything else, there's ordinal systems that out perform cardinals. There is nuance here that we need to recognize and be aware of in these discussions, especially when passion is abound. We'll also see if Clay shows up. If it stays on the front page long enough he'll probably see it (hi Clay ;) He's the real expert here.

It's perfectly possible to allow equality in ranked choice voting systems, so A > B = C > D.

But what happens in your cardinal voting system, if D and C are the front runners in competition for a single seat? Imagine the final results were

A: 30 points, B: 20 points, C: 50 points, D: 51 points.

If only you had given the realistic challenger to D your full support (5) instead of being expressive in your preference and docked them 2 points. Now D will rule for another 4 years and the C supporters will call you a traitor for voting your honest preference, just like in the present system.

And this scenario is easy to imagine. You usually know who the front-runners are, and because not only you, but everyone else, wants to avoid that scenario, belief about who the front runners are has a self-reinforcing effect.

A scenario where you're glad you didn't award the full 5 points, though, is much harder to imagine. And a scenario where you are BOTH glad that you didn't award them 5, yet glad that you didn't award them 0, is as far as I can see never going to happen in a single candidate election.

> It's perfectly possible to allow equality in ranked choice voting systems, so A > B = C > D.

While this is easy to write down it isn't easy to tabulate. You must provide a new tallying algorithm (you'll find that important features break).

> But what happens in your cardinal voting system, if D and C are the front runners in competition for a single seat?

I'm a bit confused what you're asking here. There are no voting systems where a single vote/rank permutation/rating will not change the outcome of an election. To do so would mean votes are being wasted. Yes, there are people that will be upset in these situations just like people are upset in a close sports game. But unlike a close sports game we have a higher expectation that the teams that win like this are more likely to be similar. That is candidate C's voters frequently like candidate D (your ordering even suggests that despite the example ballot being from someone on the other side of the political isle). The point is to get someone that is the most representative. In the end, it shouldn't matter what a single voter's preference is, it matters what the collective preference is. That's why we vote in the first place. So that's why I'm confused.

But maybe you should click on the montonicity link. You may also find this link[0] interesting. It is also extremely important that we remember that no voting system is perfect. So if you're going to put down a voting system because you've found a flaw you'll never be happy with a system unless you lie to yourself. It also means that if you're arguing with someone and point out a flaw you need to attempt to convince the person why that flaw (and others) are important in comparison to the upsides/downsides of the other competing system/s. Discussing voting systems is difficult because there are no global optima but it is also more difficult to discuss because proof by counter-example is meaningless and will only convince a novice. If you're wondering the motivation I'm giving in my arguments, click the links.

[0] https://web.archive.org/web/20180721221826/https://www.starv...

> While this is easy to write down it isn't easy to tabulate.

Ranked pairs etc. have no problem tabulating it. If the change to allow equal ranking breaks some obscure metric, that's a sign the metric isn't very important. After all people can just rank everyone if they worry about it.

I've read a lot about this, and I argue with the usual suspects range people every time the topic shows up.

I don't understand how you don't understand the argument. You can come into situations where you'll regret you didn't min-max your choice (rank everyone 0 or 100). You can't come into situations where you say, "gee, I'm glad I ranked that candidate somewhere in between". Consequently, everyone will do that, whether it's good for the collective or not. Sure, I want what's best for society, but I'm not volunteering to have my vote diluted for the common good - especially since I know that the people I disagree with most, certainly won't do it.

RP has a resolve time of O(n4) so I'm not sure that's a method you want to argue for under a thread where (an even lower) complexity led to a critical mistake. It would be naive to maximize VSE over all other criteria. I cannot stress enough that there are no optimal systems and that we cannot look at criteria independently.

> I don't understand how you don't understand the argument.

Because pure honesty is never an optimal strategy? I'm not aware of a system where that's the optimal strategy. But as we showed here in range, the best strategies are some form of honesty. The dishonest part is that you rescale, which I don't think most people would be upset about. What is frustrating to me is that you appear (I'm willing to admit a misread because these conversations are difficult and it is clear you're primed on my position given your usual suspects statement) to be ignoring identical strategic issues in ordinal systems. But I've also stated pretty clearly why I have a preference of cardinal systems. It is because I care a lot about the resolve complexity (the thread topic is a good example), monotonicity (something IRV/RCV fail (RP and Schultz pass but fail the prior criteria)), favorite betrayer/strong spoiler effect (again IRV/RCV fails), and an important factor is the embedding aspect. That the distance between all preference is not identical. We just can't have as richly expressive latent embedding as we can in a rated system. This has a lot of benefits that I think we need to discuss, including a smoother VSE solution space[0]. That dictates how resistant a system is to dishonest strategies and how representative we are.

[0] https://www.youtube.com/watch?v=-4FXLQoLDBA

> Because pure honesty is never an optimal strategy? I'm not aware of a system where that's the optimal strategy

It's not in a system alone. Strategies are something you have when you have information about the other players and their payoffs.

As it happens, honesty is an excellent strategy in Condorcet methods (even the "bad" ones, like those who aren't monotonic) when you have a realistic amount of information about other people's preferences/strategies. People who are honest because they can't be bothered to be strategic, do great with those methods.

In approval, even the information about who the front runners are, causes trouble.

And in range, even with NO information, you'll know it's better to just use the max and min values.

Those systems don't encourage honesty, and the "extra information" range sneaks in from self-reported strength of preferences on an absolute scale, is junk information. Nothing good can come of using it. Certainly it's no good to use it to reject a candidate that is preferred by a majority over all the other candidates.

We've been through this so many times. The original version of those simulation-based utility arguments was pushed by Warren D. Smith on RangeVoting.org.

There are so many ways to lie with simulation. When you took away even the most obvious lie in the simulations painting range in a good light - namely when you took away the simulated people's magical ability to honestly compare interpersonal utility on an absolute scale, so they always ranked their most preferred 100 and their least preferred 0, no matter where they were in "opinion space" - then most of the good properties of score vanished. Smith for a while conceded that, and advocated range with top-2 runoff if you were concerned with that (as if this was some obscure technicality).

Second, those simulations had lousy ideas of "strategy", as if strategy was just a percentage slide. Strategies are inherently dynamic, and require coordination - if I see you instructing your supporters how to vote, I can take that into account in how instruct my supporters to vote.

Yeah but I've seen how the typical person would actually do cardinal voting, because I've seen how Yelp, Uber, IMDB ratings work. Someone who's more thoughtful about it will rank each candidate (on a scale of 0-10) at reasonable values like A:3, B:5, C:7, and someone who gets all their news from tv will rank A:10, B:0, C:0. The latter is more than canceling of the reasonable person's vote.

This kind of voting rewards dumb votes over informed votes. I'd rather have ranked voting.

Yes, in ordinal ranking schemes, there's a strategy that's individually rational: Figure out who the two front runners are. Rate the better of them 100. Also, rate everyone else you like more than that front runner a 100.

If you're actually concerned about this (as far as I'm aware there's been no evidence that the voting you suggest is common when experiments have been done) you can normalize the individual ballots (e.g. softmax). But I truly believe that the added complexity to this would unnecessarily complicate an effect we do not observe. (Consider that we also have an edge case of A:10 B:10 C:10 as well. Though this ballot is easy to handle in other ways: toss it) Most voters will use a scaled sincerity voting strategy and thus you'd have A:6 B:8 C:10 and our bullet voter will no longer have an advantage. It pays to be sincere.

Continuing down this, you're essentially discussing bullet voting. I should EXPLICITLY state that this is a well known case and has been extensively studied[0][1]. I appreciate that you have concern with the system and I implore you to continue to search for flaws, but also to look for information discussing these ideas. The reason the bullet voting was likely so low in this study is that most voters are rational and understand that they are giving up significant leverage when bullet voting (note that the case is even worse in approval!). We often think our neighbors are far less rational than they actually are and we shouldn't make this mistake.

We should probably also discuss Monotonicity[2], but maybe in a follow-up conversation.

[0] https://www.rangevoting.org/BulletBugaboo

[1] https://www.rangevoting.org/RVstrat3.html (called Plurality here, strategy B)

[2] https://electionscience.org/library/monotonicity/

I don't understand how normalizing the individual ballots would change anything in the example he has given?

So a standard normalization (divide by max value) wouldn't solve much but why I specifically mentioned a softmax is because you then rescale all voting preferences to have the same power. I don't think it is a good idea btw, other than the added complexity. But it does change the above example. The problem with the example was that the first voter gave no strong preferences and thus the second voter had more power. The solution is to rescale these, which is why I said that voters tend to do this automatically (give their preferred candidate on the ballot the stop score and rate all others in proportionate to them).

I'm loving learning about these different voting methods. I do still stand by my point though.

You're right, I was arguing that voters degenerate into bullet voting over time 100% for the candidate they liked, 0% for the ones they don't. Your article [0] contends that it's not a good strategy to vote like this if you're trying to be strategic, because if candidates are A=974, B=974, F=231 and you liked F, you'd be foolish to vote 100% F, as if you like B better than A you should vote some for B.

This looks too much like the current system we have, where people don't vote for the person they like best, they cast all their votes for the person they hate the least who could beat the person they hate the most. Basically: if you normalize votes, people are going to vote 100% for the lesser of 2 evils and we're back to square 1.

If you don't normalize votes, you'll get one of the other voting strategies you've already linked to in your [1]. I expect it would essentially devolve into approval voting, as it's still voting for the lesser of 2 evils of the strongest candidates, then casting additional votes for anyone who seems viable. Perhaps it finds some other equilibrium.

Either way, I think it takes too much strategy and voter headspace compared to ranked voting, where you can just vote for the candidates you want, in the order you want, and you didn't waste a vote if your top candidate loses because your #2 vote didn't lose any weight.

I think you're trying too hard to look for errors in a system that isn't your preference and not looking hard enough at errors for the system that is your preference. All voting systems have errors and we must be clear about this.

But this strategy and consequence exists in ordinal systems just as much as cardinal systems. There is nothing stopping a voter from bullet voting on a ranked ballot. They just rank one person and no one else. The consequences are quite similar actually but might be more obvious that you're throwing away your ballot in this case. But I assure you, you're effectively throwing away your ballot in the cardinal case too.

I'm also not sure what is wrong with approval. Approval is awesome.

I am also not sure why you keep using the phrase "lesser of 2 evils". If we have a 2 candidate election then there's no point in using any of these voting systems because plurality is a strictly better system (in a 2 candidate election). In fact, it is preferential that a multi-candidate system degenerates into a plurality system when there are only two candidates! The phrase "lesser of n equals" is not really about a voting system so much as it is related to another saying: "those that seek power do not deserve it" (or Plato's alternative: "Only those who do not seek power are qualified to hold it"). This has nothing to do with a voting system but has everything to do with a candidate selection process. There's an ethical conundrum though because you need strong authoritarian power to force thrust (the potential for) power upon people. But we're digressing, just like the normalization discussion is digressing (it was intended to introduce the concept of scaled sincerity, which we see people use in practice).

Every one of your criticisms here is very valid.

> I think you're trying too hard to look for errors in a system that isn't your preference and not looking hard enough at errors for the system that is your preference.

This is true that I have that bias. I think there's currently momentum in the US for switching from single vote to ranked voting, and I worry that if we get into disputes over which alternative voting system is better, that momentum will be lost and we won't change at all. You're right I was far too quick to dismiss approval voting as a devolution from cardinal voting.

> I am also not sure why you keep using the phrase "lesser of 2 evils".

I don't know if you're from the US or not, but this is a common idiom about how our election process works in practice. The Simpsons [0], South Park [1] and dozens of others have pretty great parodies of it. In each of our presidential elections, even though there will be a plurality of presidents running, there are practically only 2 -- the ones backed by the major parties. And often, they happen to be the 2 worst -- where they're either shamelessly narcissistic enough to be unfazed by criticism, or are overly agreeable and quick to make impossible promises. Your quote by Plato is on point here.

That's the phenomenon that I desperately don't want to have any more, that I want n-candidate elections to actually have n candidates. I (perhaps wrongly) attribute a lot of the polarization in the US to two-party elections.

It's true that you can be a bullet voter on a ranked ballot, but there's no incentive to (that I can think of). In an n-candidate election, if there are 2 known popular candidates, and you absolutely despise one of them, then in a normalized cardinal system you'd be incentivized to bullet vote for the other popular candidate -- whether or not they're terrible. That's the scenario I'm talking about when I say lesser of 2 evils. Without normalization, you'd be incentivized to approval vote. And I don't have anything wrong with approval voting, I shouldn't have criticized it in my last comment.

But I do still dislike cardinal voting, because of the fact that it gives people strategies for voting. I think some people will vote earnestly like A:1, B:3, C:6 and others will vote strategically like A:10, B:0, C:0, and so this will leave the earnest voters feeling like they had the election stolen from them by strategic voters when A wins.

[0] https://www.youtube.com/watch?v=l7l9QmtiXHU

[1] https://www.youtube.com/watch?v=E7pfsneLSSM

> I worry that if we get into disputes over which alternative voting system is better, that momentum will be lost and we won't change at all

I worry that history will repeat itself and that we'll try it for awhile and then turn back because it does not resolve the things people care about (multi-parties and strong spoilers).

> I don't know if you're from the US or not

I am and demonstrated knowledge about this idiom. My comment was that the idiom is inappropriate for the conversation at hand, not that it is inappropriate for all circumstances. Why else do you think I quoted Plato?

> I want n-candidate elections to actually have n candidates.

Then you want a system that is scalable, transparent (i.e. low computational complexity, trivially parallelizable, and trivially verifiable from statistical subsets).

> It's true that you can be a bullet voter on a ranked ballot, but there's no incentive to (that I can think of).

Don't explain it to me, explain this to yourself. Why is the logic for "no incentive" invalidating in one case but not another? I argued that these are nearly identical situations and have several comments noting that there is no incentive to vote in the way you are suggesting. You are not being consistent in your logic and I'm trying to explain but it is frustrating. Maybe I am being too liberal in using my links and expecting others to read. So let me state it explicitly: there is no incentive to not give your top candidate the maximal score. So your logic is inconsistent. On the other hand, I don't expect all voters to be rational but I do expect (and have linked to experiments with people) a convergence to rationality (also consider what rational means. It doesn't mean smart but that people will play near optimally. We mix normal language and game theory language and this can create a lot of confusion).

No no your links are good, very cool to learn about rangevoting.com, I can see it has a ton to teach. But it's hard to immediately reconcile the data from these links with my own intuitions and first hand experiences and draw conclusions while still in the middle of the discussion.

You're making the argument that I shouldn't be concerned about people thresholding their votes because scaled sincerity is a nearly optimal strategy in a cardinal voting scheme, so people can vote with their heart and not feel like their vote has been wasted. That's one of the findings in the paper you shared[1] that Scaled Sincerity gets (at worst) 91% of the utility compared to Mean Based Thresholding when there are 3 candidates, 80% percent for 5 candidates, 78% for 9... etc

But there'll still inevitably be an election where 2 candidates were very close and the scaled sincerity voters who supported the 2nd place candidate feel like the election was stolen from them. The media will make an uproar about it. Voters will vote with a thresholding scheme after that. So in this case we should just skip to approval voting so people don't feel like they were robbed of the fraction of the vote they didn't contribute to their candidate.

Your other point is people will bullet vote in ranked voting and so I'm defending it too strongly. Your other article[0] very strongly argues that IRV is worse than approval voting in regards to getting bullet voters. I know it's a cop out of me to quote the criticisms that the article itself provides, but the data size really is too small. In their summary of comparisons, 16% bullet voting in Burlington and ~21% bullet voting in Irish presidential are not bad percentages in my opinion, and I don't know anything about the SF 07 or Dartmouth 06 elections.

I don't understand why the low computational complexity is important as long as the algorithm is simple to understand. Bubble sort has higher computational complexity but is a simpler algorithm than merge sort. But I don't know the details of vote counting in practice. If the computational complexity is so high that you wouldn't even determine who won until days or weeks later and was highly prone to error, then I'd stand corrected.

I don't see how cardinal voting would make a difference in there being multiple parties.

[1] https://www.rangevoting.org/RVstrat3.html

> You're making the argument that I shouldn't be concerned about people thresholding their votes because

I'm actually trying to get you to apply the same scrutiny to ordinal systems as you are giving to cardinal systems. I actually appreciate that you are critical of cardinal systems but I don't appreciate that you aren't being as critical of ordinals (recognize I'm talking about at least a dozen voting systems here. We have to start here before we can enter the nuance of individual systems but we're not even clear here). To be a good scientist you need to detach yourself from your preference. You need to critique YOUR preference more than you critique the opposing options. You're smart and if you've picked the right system then you should be able to defend it. But if you're smart and picked the wrong thing then you need to change because it isn't about what you like, it is about what is right (I'll keep stressing that no voting system is optimal and we need to define our metrics -- I have stated mine -- but we're not discussing these, we're caught up on a single strategy which appears in both cardinal and ordinal and as far as I can tell you don't recognize the latter).

> But there'll still inevitably be an election where 2 candidates were very close

Yes, and we need to think about how distant these candidates are from one another. Then ask yourself, can we compose a system where these candidates are similar? Is it always random and we have no control? Or are these candidates always distant and on opposite sides of the spectrum? You may be primed to believe the latter because that's the likely scenario in plurality voting but is this true for all systems? To answer this you need to have a deep understanding of Voter Satisfaction Efficiency (VSE), spoilers (both weak and strong -- aka favorite betrayer criteria), and how we embed preference (this is why I keep mentioning distances between candidates on the ballot. Note that this is a different distance/manifold than the previously mentioned one at the beginning of this paragraph).

> 16% bullet voting in Burlington and ~21% bullet voting in Irish presidential are not bad percentages in my opinion

I agree it is a cop-out since we see lower bullet voting tendencies in cardinal systems. I agree that sample sizes are small but we do have lots of simulations. Unfortunately we have very little real world data. Unfortunately that means we need to advocate for these systems to get real world natural experiments. Fortunately there are a few approval systems: Fargo ND and St. Luis MS (also Latvia). So we are collecting some of this data and it is very promising so far. We also have non-political versions of this like HN itself! There is only a binary voting system here (until you reach a certain score, then we get range voting). So I'd also argue we have tons of data (also surveys frequently use disapprove/neutral/approve styles of voting. Yes, this is voting). We've seen substantial success with these and I think that's not something we can exclude (though we must be aware of things like vote inflation for certain services, like Uber, that don't exhibit in other services, like old Netflix rating or current ratings).

> I don't see how cardinal voting would make a difference in there being multiple parties.

Back to the embedding.

> I could be sold on approval voting if that's the takeaway you're trying to get it.

Let's say all things are equal in effectiveness. Approval is O(n) and RCV is O(n2). I think that's killer and matters a lot (see the main topic).

> Also I don't mean to occupy either of our New Years Eves on this conversation.

This is one of the few political topics I'm extremely passionate about so I enjoy discussing it. I do want to be clear in that my goal here is not necessarily to get you to come to the cardinal side (though that'd be nice) but I want you to understand that voting methods are extremely complex (lots of people want you to think they are simple) and we shouldn't have strong passions about them without putting significant time to learn (I think discussions can be good learning mechanisms fwiw). Also plenty of time before I'm going anywhere tonight. And you have not offended me in any way. I'm glad we had this conversation.

Happy New Years :)

A cardinal voting system might be a more popular, balanced, expressive, and satisfying if you could also vote against candidates you actively dislike with negative percentages, where 0% is neutral "don't care".

So for the common case of hating all the candidates, you could vote:

A: -100% B: -100% C: -100%

There should be some way to normalize it so everyone gets the same voting power, but can choose to spend it positively or negatively in any combination.

That would give Republicans who hate Trump but refuse to vote for a Democrat the psychological out of not voting positively for any Democrats, but voting negatively for Trump.

This is an unnecessary complication but I get what you're saying. We could instead label a ballot numerically as: strongly disapprove (1), disapprove (2), neutral (3), approve (4), strongly approve (5). Or rather, both. What I'm getting at here is that nothing in the math changes so it isn't correct to say that these modifications make a system more balanced or expressive. I'll make no arguments about popularity or satisfaction since this is opinionated.

A benefit of cardinal systems that you are actually pointing out is that not only can we determine winners, but we can also get a lot of information about voter preference. This is quite useful to politicians and researchers (understanding how represented people feel). But I will also note that a strictly negative ballot is likely to be rare as it is purely a protest ballot.

Even though it may be mathematically equivalent, I think the psychological appeal is important to motivate people to vote at all, if not think clearly about how they vote, and do it honestly and wisely.

Human factors are important -- remember the Butterfly Ballots of the 2000 elections!

https://en.wikipedia.org/wiki/2000_United_States_presidentia...

I think that 0 being "don't care" and negative numbers being negative percentages is easier to grasp and more balanced and granular than (3) being neutral. Even -3 to 3 would be better than 1 to 7.

Voting "A: -100%, B: -100%, C: -100%" weights each candidate the same as "A: 0%, B: 0%, C: 0%" the same as "A: 100%, B: 100%, C: 100%", but as you said, it's useful to get an overall count of the voters' absolute satisfaction with each candidate.

A popular positively winning candidate would have a more legitimate claim to a mandate, and the lesser of two evils candidates would be branded as such.

> Even though it may be mathematically equivalent, I think the psychological appeal is important to motivate people to vote at all, if not think clearly about how they vote, and do it honestly and wisely.

I agree with you here, maybe I did not articulate my previous comment correctly to reflect this.

The biggest issue with a proposal like this is that a lack of any preference given whatsoever should not count positively towards a candidate. If someone labels candidate A as +3, candidate B as -3, and doesn't label candidate C, candidate C should not be given more preference than candidate B. People will either not understand the ramifications (they might not know someone and think that "no preference" is truly no preference when in reality it's positive preference), will forget, be lazy and not fill in some candidates, etc. I don't think that's a desirable trait since it's essentially ascribing a vote which the voter did not want. Of course if you want to make -3 (on a -3 to +3 scale) the default value assigned to "no preference" votes, perhaps that would be okay.