@sarawolk said in Allocated score (STAR-PR) centrist clones concern:

In any case, I think that Clones are a much bigger problem in hypothetical math scenarios than they ever will be in real life campaigns, and if a faction can really pull off running 2 or 3 clones that all break through and win over voters then that's frankly impressive. The reality is that if voter behavior doesn't do them in, limitations in campaign funding and volunteer power likely will.

I'm not sure I really see this as a problem of clones specifically. If parties exist, then it's fairly normal for parties to field several candidates in a multi-winner election.

But I do think the particular voting behaviour in the example election is a bit "edge case", although it's still best to avoid vulnerability to it.

While people aren't likely to cast votes that are perfectly related to utility, I still see scores as more akin to utility than to something like money, where the increase in utility drops off the higher up the scale you go.

So what I'm saying is that I see a 5 and a 0 as in the same ballpark as a 3 and a 2, rather than the 3 and 2 being preferable for equity reasons.

How good score voting is generally is a separate debate obviously, but where it gives the same tie as a pairwise method, I don't see any reason in principle to prefer one result over another. But as a tie-break, it's probably fine to choose the result you might consider less divisive.

An interesting follow-up question would be whether you would consider divisiveness over score where there isn't an exact score tie (but is a pairwise tie still) or whether it's only useful as a tie-breaker. You could, for example, reduce C's score of 3 to 2. That way, the pairwise result is still a tie but on average scores, A and B are now marginally ahead of C despite being more divisive. Is there still an argument to elect C?

Well, it partly depends on what you do with equal ranks or incomplete ballots. If an unranked candidate is scored as 0 then a 4-3-2-1 system would be different from 3-2-1-0. But if it's done in a more sensible way, they would be equivalent.

Well, aside from that, I think Arrow's Theorem is overstated in terms of its importance anyway. Pretty much all ranked-ballot methods fail IIA and if they don't they are unreasonable in some other manner. This has been known for ages anyway as it is a logical consequence of the Condorcet paradox.

So Arrow's Theorem was no great paradigm shift in our understanding. It's a non-event if you ask me.

Yeah, I mean ChatGPT has some big holes in its ability (as I've complained about), but it's also a bit scary what it's capable of sometimes. I don't think anything we can do is off limits in principle to a ChatGPT type thing.

I've been looking at this and I don't think it is the best. One (minor) problem is that when you're summing the scores, for voters that haven't had any candidates elected and also gave a score of 0 to the candidate in question, you get 0/0. Obviously you just need to count it as 0 to get it to work, but it can make one suspicious that there are problems lurking beneath.

But the main problem is that it fails scale invariance. Well it passes in a multiplicative way as it is defined on the wiki, but not if you add to the scores.

For example, if everyone scores 1 to 10 instead of 0 to 9 (so just adds 1 to every score), you can get a different result. KP + SPAV (also known as Sequential Proportional Score Voting or SPSV) passes this. I know it might seem unsatisfactory to "split" the voter with KP, but in terms of passing criteria, it seems to do the job.

@jack-waugh Right, I see. Though I think it's no better or worse than what I thought it was. Just arbitrarily favouring voters who approve a particular number of candidates. And it encourages putting up clones or non-entities accordingly.

@robla said in Paradox of Causality from Arrow’s Impossibility Theorem:

Kenneth Arrow absolutely deserved his Nobel Prize, because Arrow's Theorem was (and is) a big deal. The exact choice of criteria was beside the point; what Arrow did was describe a few "common sense" criteria and then showed them to be mutually exclusive.

But my point is that we already knew that. Amongst Arrow's criteria is IIA. And that's the one that always fails in reasonable methods. It's not like some methds fail this, others fail that. How many methods are there that are just dictatorships, for example? It's IIA all the way. Arrow's Theorem can be reworded in plain English as "With a few reasonable background assumptions, any reasonable ranked-ballot voting method fails IIA." And that has been known for centuries. Arrow just dressed it up differently.

I suppose he's used that assumption because a hereditary monarch is essentially a leader arbitrarily picked, like in random winner (as opposed to random ballot). But this is obviously very simplistic. When you have an all-powerful monarch versus some other system, the entire political and cultural landscape is likely to be very different and that isn't modelled by this.

@wolftune I'm not sure this is just an Allocated Score thing. I think that sequential methods that start by electing the candidate with the highest total score and go from there are always likely to have this sort of thing happen. Non-sequential methods may avoid it more easily, but that's obviously more expensive computationally. The other option is to not go by total score.

@cfrank I also like the concept of the Phragmén method to a point, but because it only cares about proportionality, it's not as monotonic as other methods such as Thiele's PAV method. I mean, it passes in that an extra approval can't count against a candidate, but sometimes it makes no difference, so I call it weakly monotonic.

By graph-based do you mean like Schulze for single winner? There is also a Schulze STV method, which reduces to Schulze in the single-winner case.

I think talking about Phragmén loads is over-complicating the matter. We could just say that for a method to pass (in the deterministic case), then as long as it's possible from the ballot profiles (each voter has approved enough candidates), then under any result produced by the method, it must be possible to uniquely assign each elected candidate to a voter who approved them in a way that means no voter has ≥2 candidates more than another.

For non-deterministic methods we could still talk in terms of whole candidates rather than loads and just say that the variance / total elected candidates must approach zero as the number of elected candidates increases.

@cfrank said in Proportionality criteria for approval methods:

@toby-pereira I didn’t get to check this in detail but I find it interesting. It also makes sense, although I wonder if it should be stated a bit more generally, saying maybe that there is some fixed positive constant C such that for all epsilon>0, there is some k such that for all k’>=k, C*Var(l)/k’<epsilon.

Is there a reason for choosing the normalized variance Var(l)/k’ rather than the normalized standard deviation sqrt(Var(l))/k’? Or even expressing in terms of sqrt(Var(l))/E(l)?

I could have made it specify the proportionality level more specifically, but left it more open so that it doesn't throw out non-deterministic methods such as COWPEA Lottery.

But in terms of maximum allowable load variance for a deterministic method, I think we would look at the worst case scenario of every voter approving a completely different set of candidates. We would then sequentially award a candidate to each voter until they all have one, and then start the process again. As I understand it, the variance would be highest when half the voters have an "extra" candidate and half don't.

As every candidate adds a "load" of 1, at this point every voter would have a load that is 0.5 away from the mean. So the variance would be 0.25. So instead of talking about tending towards zero, we could just say that to pass PRIL it must satisfy: var (ℓ) ≤ 0.25 (or 0.5 if we use the standard deviation instead).

I could have used other measures such as the standard deviation. I mainly used variance because of the existence of var-Phragmén. I could also have put it in terms of the leximax-Phragmén metric or Monroe.

Of course, as discussed, this doesn't define anything about the route to Perfect Representation. If there are 50 voters of party A and 50 of party B, it would allow the first 50 candidates to all go to the same party. But as also discussed, unless a method has been heavily contrived, this won't happen, and it's similar to Independence of Clones in this way.

I've been on the EndFPTP Reddit posting on this subject and I got ChatGPT to formalise the PRIL criterion. I'll copy my post from there wholesale:

On the formalisation of PRIL, I put to ChatGPT my idea of using the var-Phragmén metric and the concept of loads and told it to put the criterion in formal notation. PRIL is method agnostic though and could be used in conjunction with other methods that aim for Perfect Representation like leximax-Phragmén or Monroe. Anyway, here is what ChatGPT came up with. I make no guarantees an error hasn't slipped in:

Formal Definition of PRIL Using var-Phragmén:
Consider an election with: A set of voters N = {1, 2, . . . , n}.
A set of candidates C = {c1, c2, . . . , cm}.
Each voter i ∈ N has an approval ballot Ai ⊆ C.
A target number of winners k.
In the context of the var-Phragmén method:
Each voter i is assigned a "load" ℓi representing their share in the election of the chosen committee.

The PRIL criterion can be formalized as follows:
For any arbitrarily small positive number ϵ>0, there exists a number of winners k such that for all k′ ≥ k, the normalized variance of the voter loads satisfies:
Var (ℓ) / k′ < ϵ

This condition ensures that as the number of elected candidates k′ increases, the distribution of voter loads becomes increasingly uniform, approaching perfect representation in the limit. In essence, PRIL requires that for sufficiently large committees, the method should allocate representation so evenly among voters that the per-candidate variance of their loads becomes arbitrarily small, reflecting an ideal proportional representation.

@abel I suppose you could see the primary system as a bit of a gamble in this sense. You want to beat the other party, but you also want to do so with a candidate that is as popular as possible within the party. If your party is favourite to win anyway, you might be more inclined to just pick the most popular candidate within the party. If it's a closer election you might look at who is most likely to beat the other party. So there would be some balancing.

@bmjacobs OK, I think I get that. But I still don't think the graph is correct - it's giving a different measurement for a deterministic and non-deterministic systems. For non-deterministic methods, it seems to be working on the probability of getting elected, which I presume is what the "controlled winning probability" on the y axis means. But for the deterministic methods, it just gives a score of 100 if they are guaranteed to be elected and 0 otherwise.

If it's a about guaranteed election then the non-deterministic methods should give 0 across the board. If it's probabilistic, then the non-deterministic methods are correct on the graph but the deterministic ones are wrong - though there isn't an exact probability you can give as it depends on voting patterns.

@bmjacobs Interesting article, but a couple of things. Where does it come from that in Borda you need 2/3 of the vote to ensure a win? Also Borda and winner-takes-all systems wouldn't have the step function if there are more than two candidates. E.g. having 40% of the vote under First Past the Post will still give you the win in many elections. If it's to guarantee the win then yes, but I don't think it's clear that that's what the graph is trying to say.

@mkeypaige A lot of those topics have Wikipedia or Electowiki pages, which describe them quite well, and there are also YouTube videos which discuss a lot of them. Some of the later stuff I wouldn't even know what they are. But I'll give you some links.

Ranked voting - Wikipedia

1. Plurality system (also known as First Past the Post) - Wikipedia, CGP Grey video

2. Ranked Choice Voting (also known as Instant Runoff Voting, Alternative Vote among others): Wikipedia, CGP Grey video

3. Condorcet methods: Wikipedia, Carneades.org video

4. Borda Count: Wikipedia, Carneades.org video

5. Majority criterion: Wikipedia, Becky Moening video

6. Unanimity criterion: Wikipedia article on Pareto Efficiency, Eric Pacuit video

7. Condorcet winner criterion: Wikipedia, Carneades.org video

8. Monotonicity criterion: Electowiki article, Becky Moening video

9. IIA (Independence of Irrelevant Alternatives) criterion: Wikipedia, Carneades.org video

10. Manipulability: Wikipedia article on strategic voting, Katherine Heller video

11. Random dictator: Wikipedia article, Carneades.org video

12. Approval voting: Wikipedia, CGP Grey video

That's the first section covered. Let me know if this is useful at all and I can get some links for the others.

@cfrank said in Nonmonotonic methods are unconstitutional in Germany?:

What I mean is, it should not be the case that two rational voters with equal access to information, and who cast the exact same ballot, nevertheless conflict in their expectations about how that ballot will contribute to the electoral result. Non-monotonicity introduces unnecessary conflicts of interest. So I really doubt that under sufficient scrutiny, any non-monotonic system would actually stand up to the phrasing of the law.
Whether or not they are unconstitutional in Germany or elsewhere, I do believe that any voting system that is not monotonic or that fails independence of clones should be considered fully improper and disqualified from being part of any official public elections. I.e. they should by all reason be illegal in that context. And perhaps actually are in spirit or will be found formally illegal in Germany.

I consider failing participation to be on the same level as failing monotonicity. Philosophically I'd say they are very similar, even if they tend to happen for different mathematical reasons. But because participation is a more "expensive" criterion, people tend to be less harsh on methods that fail it.

I've said previously on this group that non-deterministic elections can be a good way to simplify proportional representation. It doesn't have to be as crude as simple random ballot where you have one representative per constituency selected by the drawing of a single ballot. As I said here just earlier this month:

Proportional methods tend to just be more complex by their nature. But if you allow them to be non-deterministic then that goes away. E.g. COWPEA Lottery which uses approval ballots. Or if you have a region that elects, say, 6 candidates, voters just rank their top 6 candidates. Then you consecutively pick six ballots at random and elect the unelected candidate that is highest ranked on that ballot.

This type of method, while it doesn't guarantee a very proportional result in each region, would actually give better proportionality nationally than deterministic methods that use these smallish regions (like STV), and they also keep the election candidate-based, which other nationally proportional methods tend not to.

Random ballot with just one representative per region guarantees that honest voting is the best strategy, but I tend to think that it becomes too lotteristic at that point. With e.g. five or six chances to be elected (as in the above methods), particularly popular candidates would not be on such a knife-edge of being elected.

I also think that non-deterministic methods send out a good message - that there are no "safe seats", and that representing the electorate is a privilege and not some guaranteed right.

So while non-deterministic methods might be a tough sell, I personally prefer them for national parliaments.