@marylander said in Rule X extended to score ballots:

On the other hand, RRV and STV choose winner sets where all voters are strictly worse off than under the SSS winner set,

It's weird that RRV has done that since its mechanism is just to maximise the "satisfaction" score for each voter. I presume then that this is to do with electing sequentially rather than something fundamental to RRV itself. And I would also presume that electing sequentially can throw out weird anomalies for any voting method, and I don't see any particular reason why any method should be more susceptible than any other method to this.

As an aside, regardless of what one thinks of Thiele methods in general, I do not consider RRV to be a good implementation of it.

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.

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.

@marylander said in Rule X extended to score ballots:

@toby-pereira Do you think that EJR in particular is clumsy, or all of them in general?

Yes, because as I said in the archived discussion - "all these criteria seem oddly weak in that they only make demands about a single voter, or in proportional justified representation just one voter per candidate that the group "deserves"."

I don't have any specific examples in mind to create a clone failure, but if you have an A>B>C>A cycle, then by cloning one of the candidates, you can award the win to the candidate that pairwise beats that candidate. Clone A and C wins; clone B and A wins; clone C and B wins. So to make something up on the fly (not necessarily realistic):

14: A>B>C
10: B>C>A
5: C>A>B

In this example, A beats B 19 to 10; B beats C 24 to 5 and C beats A 15 to 14. C has the worst pairwise win and the worst pairwise loss. But clone A and C will win. (Sorry for not going into Codepens.)

@rob said in STAR-like method ("reverse STAR"?):

So.... in a general sense, whether were talking about voting methods, seat belts, motorcycle helmets, or vaccines (sorry! political! ), I'm far more interested in knowing the degree of protection, as opposed to the simple boolean answer to whether or not it provides 100% protection. In my opinion, the latter is rarely useful, and often destructive to the goal of making any improvements at all.

My current feeling on it is that the benefit this method has in simplicity and ease of explaining far outweighs any risk of clones causing a problem. I'm aware that Schultz and ranked pairs seem to solve this, but they do it at the cost of complexity which, in my opinion, is simply unmarketable. Meanwhile STAR seems easier to sell than Schultz or ranked pairs, but if there is ever an election where there is a Condorcet winner, and STAR chooses a different candidate, I would consider that a serious flaw... partly because it is directly observable that this happened, after the fact. (a la Burlington 2009 with IRV) There would be no end to people claiming it elected the wrong candidate.

I agree with some of this. It seems to me that you want to find the simplest Condorcet method, since in terms of results, there won't be much between them.

So then the question becomes - is this the simplest Condorcet method? I don't know but it's probably quite good in that respect. I don't actually think ranked pairs is complex to understand, although I think Schulze is. Also there's the method that I think is called "Benham" where you sequentially eliminate the candidate with the fewest first places until a Condorcet winner exists. Though you don't have to use the word "Condorcet" in describing it. Just a candidate that head-to-head beats all the others.

There are of course cases where it's debatable whether you'd want the Condorcet winner to be elected. E.g.

49: A>>C>B
49: B>>C>A
2: C

Basically, two mainstream polarising candidates and a non-entity who is the Condorcet winner.

@brozai I suppose I find them both clumsy and weak. Take Justified Representation. It refers to the case where a Hare quota of voters all approve one candidate. Intuitively, you might think that as a solid Hare quota, they should all get this candidate elected. But further reflection suggests this isn't always possible.

For example, there might be two candidates to be elected in an election. Three candidates are approved by 50%, 51% and 51% of the electorate respectively with no correlation between voting for one or any of the others. In this case the two on 51% would be elected, so not the candidate on 50%.

Of the voters who approved the 50% candidate (it has a full Hare quota), slightly under a quarter of them would not have a candidate elected.

It would be impossible to guarantee that every voter in a Hare quota group gets a candidate elected. So how do we make a criterion around it? Well, they went for the minimum possible option - one of the voters must have a candidate elected. At that stage it just seems like a pointless nod towards something they were probably trying to achieve but realised they had to back out of.

I've just been looking through old threads and found this one.

@nevinbr said in Least-bad Single-winner Ranking Method?:

Objectively, the answer is Game Theory Voting. However, it is basically impossible to explain to, well, anyone.

This method was also discussed at greater length on the old CES Google Group here. But the point is that while it is optimal in the sense that you might want to optimise this one particular thing, you might not want to optimise that one particular thing, so it would be wrong to say that it's objectively the best method.

In fact, when you look at what it is trying to achieve, it becomes quite clear that it's really just a bit of game theory fun rather than a method seriously trying to optimise something that a human would want to optimise. And you can see that by an example that you gave in the CES thread.

To expand on that last point, consider the following election with 3 candidates and 100 voters:

49 ABC

48 CAB

3 BCA

This has a Condorcet cycle:

A beats B by 94

B beats C by 4

C beats A by 2

At first glance, we notice that A has the largest victory and the smallest defeat, as well as the highest Borda total. However, the GT method elects A just 4% of the time.

Can that really be optimal?

Let’s think about it.

Okay, B gets crushed by A and barely squeaks by C. We don’t want to get crushed because that is bad for our long-term average, so B probably should not win. And if B loses (or didn’t run at all) then C defeats A.

It seems that A’s victory over B is mostly ephemeral. As much as we would like to score that +94 to improve our long-term average, the only way it happens is if we pick A and the system we’re up against picks B. But since B shouldn’t win, we expect the other system won’t pick B either.

In particular, if we usually pick A and the other system usually picks C, then we are going to lose frequently. We would rather be that other system and pick C most often, which is exactly what GT does. The optimal distribution is:

A wins 4%

B wins 2%

C wins 94%

Moreover, in any 3-way Condorcet cycle, the probability of each candidate winning is always proportional to the margin of victory between the other two candidates. And this is provably optimal.

In the example ballots, it seems clear that A is the best winner. However, under this method, because B definitely isn't a good winner, other methods won't select B so the A>B pairwise win might as well be ignored. Better concentrate on A v C instead and since C wins that, C is the overall best pick.

So what this method does is not pick the candidate that is somehow judged to be best for society, but the candidate that has the best average margin of victory against a candidate picked by another method playing this same game. And this method would never lose on average against another method (though obviously might tie - e.g. against itself). I don't see how this optimality relates to real life at all and why it would be good for us to adopt it.

To make this clear, the ranked pairs method is a Condorcet method that a lot of people like. But I could devise a method that elects a candidate that pairwise beats the ranked pairs winner whenever one exists. Otherwise elect the ranked pairs winner. According the metric used here, this method is better than ranked pairs - when these two methods are viewed as the only choices at least. But is this method better tban ranked pairs by any reasonable measure? Of course not.

An interesting academic exercise, but nothing more. Certainly not objectively the best single-winner voting method.

@jack-waugh Are Jack Waugh and William Waugh the same person? It's something I've often wondered.

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.

I've just been looking through old threads and found this one.

@nevinbr said in Least-bad Single-winner Ranking Method?:

Objectively, the answer is Game Theory Voting. However, it is basically impossible to explain to, well, anyone.

This method was also discussed at greater length on the old CES Google Group here. But the point is that while it is optimal in the sense that you might want to optimise this one particular thing, you might not want to optimise that one particular thing, so it would be wrong to say that it's objectively the best method.

In fact, when you look at what it is trying to achieve, it becomes quite clear that it's really just a bit of game theory fun rather than a method seriously trying to optimise something that a human would want to optimise. And you can see that by an example that you gave in the CES thread.

To expand on that last point, consider the following election with 3 candidates and 100 voters:

49 ABC

48 CAB

3 BCA

This has a Condorcet cycle:

A beats B by 94

B beats C by 4

C beats A by 2

At first glance, we notice that A has the largest victory and the smallest defeat, as well as the highest Borda total. However, the GT method elects A just 4% of the time.

Can that really be optimal?

Let’s think about it.

Okay, B gets crushed by A and barely squeaks by C. We don’t want to get crushed because that is bad for our long-term average, so B probably should not win. And if B loses (or didn’t run at all) then C defeats A.

It seems that A’s victory over B is mostly ephemeral. As much as we would like to score that +94 to improve our long-term average, the only way it happens is if we pick A and the system we’re up against picks B. But since B shouldn’t win, we expect the other system won’t pick B either.

In particular, if we usually pick A and the other system usually picks C, then we are going to lose frequently. We would rather be that other system and pick C most often, which is exactly what GT does. The optimal distribution is:

A wins 4%

B wins 2%

C wins 94%

Moreover, in any 3-way Condorcet cycle, the probability of each candidate winning is always proportional to the margin of victory between the other two candidates. And this is provably optimal.

In the example ballots, it seems clear that A is the best winner. However, under this method, because B definitely isn't a good winner, other methods won't select B so the A>B pairwise win might as well be ignored. Better concentrate on A v C instead and since C wins that, C is the overall best pick.

So what this method does is not pick the candidate that is somehow judged to be best for society, but the candidate that has the best average margin of victory against a candidate picked by another method playing this same game. And this method would never lose on average against another method (though obviously might tie - e.g. against itself). I don't see how this optimality relates to real life at all and why it would be good for us to adopt it.

To make this clear, the ranked pairs method is a Condorcet method that a lot of people like. But I could devise a method that elects a candidate that pairwise beats the ranked pairs winner whenever one exists. Otherwise elect the ranked pairs winner. According the metric used here, this method is better than ranked pairs - when these two methods are viewed as the only choices at least. But is this method better tban ranked pairs by any reasonable measure? Of course not.

An interesting academic exercise, but nothing more. Certainly not objectively the best single-winner voting method.

One criterion that I would regard as essential for any approval-based proportional method is that in the case where the number of candidates equals the number of voters, then if the ballots make it possible for each voter to be uniquely assigned a candidate that they approved, then such a result must be the result.

It's a bit like perfect representation, except that perfect representation makes the demand for fractions of candidates. (E.g. if it's possible for each voter to be assigned their own unique 0.05 of a candidate - allowing e.g. for 0.025 each for two candidates - then this must happen.)

A similar criterion would be that for each new candidate you add to the committee from 1 upwards, they must be assignable to a different voter, until every voter has their own candidate (assuming the ballots make it possible). It would then continue with each voter getting a second candidate in turn if even more candidates were added.

Thiele-based methods fail this.

@keith-edmonds said in Rule X extended to score ballots:

Consider this 5 winner example with clones for each candidate
Red: 61% vote A:5, B:3, C:0
Blue: 39% vote A:0, B:3, C:5

RRV Gives ['A1', 'C1', 'A2', 'B1', 'B2']
MES Gives ['A1', 'A2', 'A3', 'C1', 'B1']
SSS Gives ['A1', 'B1', 'B2', 'B3', 'B4']
Allocated score Gives ['A1', 'B1', 'A2', 'B2', 'A3']
STV Gives ['A1', 'A2', 'A3', 'C1', 'C2']

I could have made a calculational error but I did it with code which I can post if people want to look for bugs. If correct this is super interesting. They all give different results.

Which sets are in the core? If any?

Just out of interest, I worked this out with COWPEA + KP and got the following percentages (assuming I calculated correctly):

A: 43.1%
B: 34.3%
C : 22.6%

This would probably mean 2 As, 2Bs and a C in a five-seat constituency (the RRV result). Which I think you consider to be not a great result.

@keith-edmonds said in Rule X extended to score ballots:

I would not say that. Systems like SSS and MES are designed to be sequential. That may be the issue with RRV but SSS and MES do not have the same excuse. DSV is the sequential implementation of Thiele for score. Or at least I designed it to be a close to SPAV as I could.

When you gave the results for your example in the post a few above you said RRV, but do you mean that you actually used SDV? Edit - In any case I don't think RRV is the method worth calculating results for.

@toby-pereira said in Rule X extended to score ballots:

As an aside, regardless of what one thinks of Thiele methods in general, I do not consider RRV to be a good implementation of it.

You prefer Sequential Proportional Score Voting, correct?

Yeah, that's the one that's SPAV + KP, right? I think that's still my preferred Thiele-based option. But either that or SDV are likely superior to RRV.

@marylander said in Rule X extended to score ballots:

On the other hand, RRV and STV choose winner sets where all voters are strictly worse off than under the SSS winner set,

It's weird that RRV has done that since its mechanism is just to maximise the "satisfaction" score for each voter. I presume then that this is to do with electing sequentially rather than something fundamental to RRV itself. And I would also presume that electing sequentially can throw out weird anomalies for any voting method, and I don't see any particular reason why any method should be more susceptible than any other method to this.

As an aside, regardless of what one thinks of Thiele methods in general, I do not consider RRV to be a good implementation of it.

@marylander said in Rule X extended to score ballots:

On the other hand, RRV and STV choose winner sets where all voters are strictly worse off than under the SSS winner set, so if we make the assumption* that the sum of the scores can be used to determine which overall committee the voter would approve, then could be interpreted as quite a bad example for RRV and STV.

* I'm not calling it an unreasonable assumption, but it is an assumption and so I'm stating it. Perhaps we could test it with surveys, although in my opinion the meaning of scores depends on the voting system to some extent, so it might not be easy.

Even aside from scores, and looking at full approvals, there are scenarios where a "Pareto dominated" result is arguably better. See the archive here.

@andy-dienes said in Rule X extended to score ballots:

@toby-pereira I think this example actually shows my point. In this case, (with very high probability if the approval sets are truly uncorrelated) any two of the winners will satisfy EJR (and thus also PJR/JR), so it is not restrictive at all.

My point wasn't that it was restrictive, but that it seemed a bit weak, making requirements about just one voter (but you acknowledged that later in your post).

Unless I'm misunderstanding the setup, I'm not sure why you are saying the two 51% will necessarily be elected (although, in this case it does seem like the 'right' choice).

Well, the two 51% candidates should be elected under any reasonable method given the lack of any correlation. But in any case, the point is that it shows that approximately 1/8 of the electorate will be unrepresented despite being part of a Hare quota.

Edit: I should probably mention there is another intuitive criterion, perfect representation. This is when the voters can be exactly divided into quotas such that each quota gets a unanimous winner. Obviously, this is also not always possible, but more importantly it is incompatible with EJR. This is one reason maybe it's reasonable to consider EJR 'clumsy.' However, it is compatible with PJR. It seems to me that PJR is weak enough such that any noncompliance is likely indicative of deeper problems. In particular, optimization of the maxPhragmen metric implies PJR. Your 'squared load' metric I believe is equivalent to the varPhragmen objective function, which implies JR.

There are cases where it is arguably undesirable to have perfect representation. I added an example to the wiki page. But to copy and paste:

Consider the following election with two winners, where A, B, C and D are candidates, and the number of voters approving each candidate are as follows:

100 voters: A, B, C

100 voters: A, B, D

1 voter: C

1 voter: D

A method passing the perfect representation criterion must elect candidates C and D despite near universal support for candidates A and B. This could be seen as an argument against perfect representation as a useful criterion.

Also, on PJR, it's worth pointing out that Sainte-Laguë/Webster can in some circumstances fail the lower quota rule, so presumably fails this criterion. See example on Warren Smith's site. And I would generally consider this to be a fair system.

@brozai I suppose I find them both clumsy and weak. Take Justified Representation. It refers to the case where a Hare quota of voters all approve one candidate. Intuitively, you might think that as a solid Hare quota, they should all get this candidate elected. But further reflection suggests this isn't always possible.

For example, there might be two candidates to be elected in an election. Three candidates are approved by 50%, 51% and 51% of the electorate respectively with no correlation between voting for one or any of the others. In this case the two on 51% would be elected, so not the candidate on 50%.

Of the voters who approved the 50% candidate (it has a full Hare quota), slightly under a quarter of them would not have a candidate elected.

It would be impossible to guarantee that every voter in a Hare quota group gets a candidate elected. So how do we make a criterion around it? Well, they went for the minimum possible option - one of the voters must have a candidate elected. At that stage it just seems like a pointless nod towards something they were probably trying to achieve but realised they had to back out of.