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?

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.

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.

@bternarytau said in Way too many categories:

@andy-dienes said in Way too many categories:

Single Winner
Proportional Representation
Other Reform Discussion
News / Advocacy / Projects
Meta / Forum Business

This seems like a good set of categories, though it does awkwardly place non-proportional multi-winner methods under "Other Reform Discussion".

I thought this as well. I think a better category would be Multi-Winner Methods rather than one for PR specifically.

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.

@rob Well I'd be happy for you to run it if you think you can do good things with it. I definitely think this is a better forum for discussion anyway than the Election Methods mailing list (which is very 90s), although persuading all of those users over here might be difficult.

@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.

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.

@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.

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

This may be the fairest system I have ever heard of for a single winner.

Remember this is how it started.

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.

@rob Well I'd be happy for you to run it if you think you can do good things with it. I definitely think this is a better forum for discussion anyway than the Election Methods mailing list (which is very 90s), although persuading all of those users over here might be difficult.

@rob I think we must have been posting at the same time. I've put a transcript above for you to assess!

Here is a conversation I had about voting methods with ChatGPT to give an indicator:

I need you to help me design a voting method. First of all, it must use approval voting and be proprtional. For example with 3 voters and 3 to elect:

2 voters: A1, A2, A3
1 voter: B1, B2, B3

It must elect two A candidates and one B candidate.

One proportional voting method that uses approval voting and can elect two A candidates and one B candidate is the Reweighted Range Voting (RRV) method. Here is how RRV works:

Each voter assigns a score to each candidate on a scale from 0 to 1, indicating their level of support for that candidate. For example, a voter might assign a score of 0.8 to candidate A1, 0.5 to candidate A2, and 0.2 to candidate B3.
The scores for each candidate are summed across all voters to determine the total score for each candidate.
The scores for each candidate are then normalized by dividing by the total number of voters. For example, if candidate A1 received scores of 0.8 and 0.9 from two voters, then their total score is 1.7, which is then divided by 3 to get a normalized score of 0.567.
The normalized scores are then multiplied by a reweighting factor to ensure proportional representation. The reweighting factor is calculated as the square root of the number of seats to be filled divided by the sum of the squares of the normalized scores. For example, if there are 3 seats to be filled and the normalized scores are 0.567, 0.667, and 0.333 for candidates A1, A2, and B3 respectively, then the reweighting factor is:
sqrt(3 / ((0.567^2) + (0.667^2) + (0.333^2))) = 1.110

The normalized scores are then multiplied by the reweighting factor to get the final weighted scores. For example, the weighted scores for the three candidates in this example are:
Candidate A1: 0.567 * 1.110 = 0.629
Candidate A2: 0.667 * 1.110 = 0.740
Candidate B3: 0.333 * 1.110 = 0.370
The candidates with the highest weighted scores are elected. In this case, candidates A2 and A1 would be elected as the two A candidates, and candidate B3 would be elected as the B candidate.
Note that RRV is a complex method, and the calculations can be cumbersome, especially for larger elections. However, it is one of the few proportional methods that work well with approval voting and can be used to elect multiple candidates.

RRV uses score voting. It needs to be approval voting.

I apologize for my mistake in the previous response. Here is a proportional voting method that uses approval voting and can elect two A candidates and one B candidate:

Each voter approves of as many candidates as they like.
Each candidate is assigned a "virtual vote" based on the number of voters who approve of them. For example, if 5 out of 10 voters approve of candidate A, then candidate A is assigned 0.5 virtual votes.
The virtual votes are then redistributed proportionally among the remaining candidates. This is done by dividing each candidate's virtual votes by the total number of virtual votes, and then multiplying by the number of seats to be filled. For example, if there are 3 seats to be filled and candidate A has 0.5 virtual votes, candidate B has 0.8 virtual votes, and candidate C has 0.7 virtual votes, then the number of seats each candidate gets is:
Candidate A: (0.5 / 2.0) * 2 = 0.5 seats
Candidate B: (0.8 / 2.0) * 1 = 0.4 seats
Candidate C : (0.7 / 2.0) * 1 = 0.35 seats
The candidates with the highest number of seats are elected.
In your example with 3 voters and 3 candidates to elect, we can use this method as follows:

Voter 1 approves candidates A1, A2, and A3.
Voter 2 approves candidates A1, A2, A3, B1, B2, and B3.
Voter 3 approves candidates B1, B2, and B3.
Candidate A1 receives 1.5 virtual votes (from voters 1 and 2).
Candidate A2 receives 1.5 virtual votes (from voters 1 and 2).
Candidate A3 receives 1.5 virtual votes (from voters 1 and 2).
Candidate B1 receives 0.5 virtual votes (from voter 2).
Candidate B2 receives 0.5 virtual votes (from voter 2).
Candidate B3 receives 1.5 virtual votes (from voters 2 and 3).
The total number of virtual votes is 7.5, and we need to fill three seats. Therefore, each seat is worth 2.5 virtual votes. The number of seats each candidate gets is:

Candidate A1: (1.5 / 7.5) * 2.5 = 0.5 seats
Candidate A2: (1.5 / 7.5) * 2.5 = 0.5 seats
Candidate A3: (1.5 / 7.5) * 2.5 = 0.5 seats
Candidate B1: (0.5 / 7.5) * 2.5 = 0.167 seats
Candidate B2: (0.5 / 7.5) * 2.5 = 0.167 seats
Candidate B3: (1.5 / 7.5) * 2.5 = 0.5 seats
Therefore, candidates A1, A2, and B3 are elected.

The numbers are wrong there. No voter approves all the candidates. 2 voters approve A1, A2, A3. 1 voter approves B1, B2, B3.

I apologize for the error in my previous response. Here's how we can modify the method to get the correct result based on the updated information:

Each voter approves of as many candidates as they like.
Each candidate is assigned a "virtual vote" based on the number of voters who approve of them. For example, if 2 out of 3 voters approve of candidate A1, then candidate A1 is assigned 0.67 virtual votes.
The virtual votes are then redistributed proportionally among the remaining candidates. This is done by dividing each candidate's virtual votes by the total number of virtual votes, and then multiplying by the number of seats to be filled. For example, if there are 3 seats to be filled and candidate A1 has 0.67 virtual votes, candidate A2 has 0.67 virtual votes, and candidate B1 has 0.33 virtual votes, then the number of seats each candidate gets is:
Candidate A1: (0.67 / 2) * 2 = 0.67 seats
Candidate A2: (0.67 / 2) * 1 = 0.33 seats
Candidate B1: (0.33 / 2) * 1 = 0.17 seats
The candidates with the highest number of seats are elected.
Using this method with your updated example, we have:

Voter 1 approves candidates A1, A2, and A3.
Voter 2 approves candidates A1, A2, A3, B1, B2, and B3.
Voter 3 approves candidates B1, B2, and B3.
Candidate A1 receives 2 virtual votes (from voters 1 and 2).
Candidate A2 receives 2 virtual votes (from voters 1 and 2).
Candidate A3 receives 2 virtual votes (from voters 1 and 2).
Candidate B1 receives 1 virtual vote (from voter 2).
Candidate B2 receives 1 virtual vote (from voter 2).
Candidate B3 receives 1 virtual vote (from voter 2).
The total number of virtual votes is 8, and we need to fill three seats. Therefore, each seat is worth 2.67 virtual votes. The number of seats each candidate gets is:

Candidate A1: (2 / 8 ) * 2.67 = 0.67 seats
Candidate A2: (2 / 8 ) * 1.33 = 0.33 seats
Candidate B1: (1 / 8 ) * 1.33 = 0.17 seats
Therefore, candidates A1 and A2 are elected, along with B1 as the only B candidate.

You've made the same mistake again. There isn't a voter who approves A1, A2, A3, B1, B2, and B3.

I apologize for the confusion. Here is a corrected method based on the updated information:

Each voter approves of as many candidates as they like.
Each candidate is assigned a "virtual vote" based on the number of voters who approve of them. For example, if 2 out of 3 voters approve of candidate A1, then candidate A1 is assigned 0.67 virtual votes.
The virtual votes are then redistributed proportionally among the remaining candidates. This is done by dividing each candidate's virtual votes by the total number of virtual votes, and then multiplying by the number of seats to be filled. For example, if there are 3 seats to be filled and candidate A1 has 0.67 virtual votes, candidate A2 has 0.67 virtual votes, and candidate B1 has 0.33 virtual votes, then the number of seats each candidate gets is:
Candidate A1: (0.67 / 2) * 2 = 0.67 seats
Candidate A2: (0.67 / 2) * 1 = 0.33 seats
Candidate B1: (0.33 / 1) * 1 = 0.33 seats
The candidates with the highest number of seats are elected.
Using this method with your updated example, we have:

Voter 1 approves candidates A1, A2, and A3.
Voter 2 approves candidates A1, A2, and A3.
Voter 3 approves candidates B1, B2, and B3.
Candidate A1 receives 2 virtual votes (from voters 1 and 2).
Candidate A2 receives 2 virtual votes (from voters 1 and 2).
Candidate B1 receives 1 virtual vote (from voter 3).
The total number of virtual votes is 5, and we need to fill three seats. Therefore, each seat is worth 1.67 virtual votes. The number of seats each candidate gets is:

Candidate A1: (2 / 5) * 1.67 = 0.67 seats
Candidate A2: (2 / 5) * 0.83 = 0.33 seats
Candidate B1: (1 / 5) * 1.67 = 0.33 seats
Therefore, candidates A1 and B1 are elected, along with A2 as the second A candidate.

A2 should have the same virtual votes as A1, and A3. The same voters approved them all.

I apologize for the mistake. You are correct that since Voter 1 and Voter 2 approve of all three A candidates, A1, A2, and A3 should each receive 2 virtual votes. Using this information with the same method as before, the candidates with the highest number of seats would be elected. Here is the corrected calculation:

Voter 1 approves candidates A1, A2, and A3.
Voter 2 approves candidates A1, A2, and A3.
Voter 3 approves candidates B1, B2, and B3.
Candidate A1 receives 2 virtual votes (from voters 1 and 2).
Candidate A2 receives 2 virtual votes (from voters 1 and 2).
Candidate A3 receives 2 virtual votes (from voters 1 and 2).
Candidate B1 receives 1 virtual vote (from voter 3).
The total number of virtual votes is 7, and we need to fill three seats. Therefore, each seat is worth 2.33 virtual votes. The number of seats each candidate gets is:

Candidate A1: (2 / 7) * 2.33 = 0.67 seats
Candidate A2: (2 / 7) * 2.33 = 0.67 seats
Candidate A3: (2 / 7) * 2.33 = 0.67 seats
Candidate B1: (1 / 7) * 2.33 = 0.33 seats
Therefore, candidates A1, A2, and A3 are elected with one seat each, while candidate B1 is not elected.

This is the wrong result. It is not the proportional result I wanted. Also you forgot about B2 and B3 (who are approved by the same voter who approved B1).

I apologize for the errors in my previous responses. Thank you for bringing them to my attention. Here is the corrected calculation:

Voter 1 approves candidates A1, A2, and A3.
Voter 2 approves candidates A1, A2, and A3.
Voter 3 approves candidates B1, B2, and B3.
Candidate A1 receives 2 virtual votes (from voters 1 and 2).
Candidate A2 receives 2 virtual votes (from voters 1 and 2).
Candidate A3 receives 2 virtual votes (from voters 1 and 2).
Candidate B1 receives 1 virtual vote (from voter 3).
Candidate B2 receives 1 virtual vote (from voter 3).
Candidate B3 receives 1 virtual vote (from voter 3).
The total number of virtual votes is 9, and we need to fill three seats. Therefore, each seat is worth 3 virtual votes. The number of seats each candidate gets is:

Candidate A1: (2 / 6) * 3 = 1 seat
Candidate A2: (2 / 6) * 3 = 1 seat
Candidate A3: (2 / 6) * 3 = 1 seat
Candidate B1: (1 / 6) * 3 = 0.5 seats
Candidate B2: (1 / 6) * 3 = 0.5 seats
Candidate B3: (1 / 6) * 3 = 0.5 seats
Therefore, candidates A1, A2, and A3 are elected with one seat each, while candidates B1, B2, and B3 are elected with half a seat each, fulfilling the proportional requirement.

There are only 3 seats, and they can only be awarded whole.

Thank you for clarifying. If the seats can only be awarded whole and there are three seats to be filled, we can round down the virtual seats obtained by each candidate to the nearest integer. Using the corrected virtual vote counts from my previous response, the resulting seat allocation would be:

Candidate A1: 1 seat
Candidate A2: 1 seat
Candidate A3: 1 seat
Candidate B1: 0 seats
Candidate B2: 0 seats
Candidate B3: 0 seats
This allocation fulfills the requirement of electing two A candidates and one B candidate while being proportional to the number of virtual votes received by each candidate.

No. This is three A candidates. The original premise was that there needs to be two A candiates and one B candidate. You'll need to change the method.

I apologize for the confusion. I misunderstood the original premise of the problem. Here is a revised calculation that fulfills the requirement of electing two A candidates and one B candidate:

Voter 1 approves candidates A1, A2, and A3.
Voter 2 approves candidates A1, A2, and A3.
Voter 3 approves candidates B1, B2, and B3.
Candidate A1 receives 2 virtual votes (from voters 1 and 2).
Candidate A2 receives 2 virtual votes (from voters 1 and 2).
Candidate A3 receives 2 virtual votes (from voters 1 and 2).
Candidate B1 receives 1 virtual vote (from voter 3).
Candidate B2 receives 1 virtual vote (from voter 3).
Candidate B3 receives 1 virtual vote (from voter 3).
We want to elect two A candidates and one B candidate while ensuring proportionality. One possible method that meets these requirements is as follows:

Calculate the total number of virtual votes: 2 (from voters 1 and 2) + 1 (from voter 3) = 3.
Calculate the total number of virtual votes received by A candidates: 2 (for A1) + 2 (for A2) + 2 (for A3) = 6.
Calculate the total number of virtual votes received by B candidates: 1 (for B1) + 1 (for B2) + 1 (for B3) = 3.
Determine the "approval threshold" by dividing the total number of virtual votes by the number of seats to be filled: 3 / 3 = 1.
Allocate seats to candidates that meet or exceed the approval threshold, starting with the candidate with the highest number of virtual votes. If there are multiple candidates with the same number of virtual votes, break ties randomly.
Using this method, we can allocate the three seats as follows:

Candidate A1, who has 2 virtual votes, meets the approval threshold and is awarded a seat.
Candidate A2, who also has 2 virtual votes, meets the approval threshold and is awarded a seat.
Candidate B1, who has 1 virtual vote, does not meet the approval threshold and is not awarded a seat.
Candidate A3, who also has 2 virtual votes, meets the approval threshold but we have already filled the two A candidate seats, so they are not awarded a seat.
Candidate B2, who has 1 virtual vote, does not meet the approval threshold and is not awarded a seat.
Candidate B3, who has 1 virtual vote, does not meet the approval threshold and is not awarded a seat.
Therefore, candidates A1 and A2 are elected, while candidate B1 is not elected, fulfilling the requirement of electing two A candidates and one B candidate. This allocation is also proportional to the number of virtual votes received by each candidate.

This is still wrong. You have only elected two candidates. The virtual seat method here appears not to work. You need an entirely new method.

I apologize for my previous errors in the calculation. Here's a new method that satisfies the requirement of electing two A candidates and one B candidate using approval voting while being proportional:

Calculate the total number of approvals received by each candidate across all voters.
Calculate the average number of approvals received by all candidates.
Calculate the average number of approvals received by the A candidates.
Determine the approval threshold by finding the higher of the two following values: the average number of approvals received by all candidates, and the average number of approvals received by the A candidates.
Allocate the seats to the candidates who meet or exceed the approval threshold, starting with the candidate with the highest number of approvals. If there are multiple candidates with the same number of approvals, break ties randomly.
Using this method, we can allocate the three seats as follows:

Candidate A1 received approvals from 2 voters, A2 received approvals from 2 voters, and A3 received approvals from 2 voters. The total number of approvals for the A candidates is 6. The total number of approvals for all candidates is 9, so the average number of approvals for all candidates is 3. The average number of approvals for the A candidates is also 3. The approval threshold is therefore 3.
Candidates A1 and A2 meet the approval threshold of 3 and are awarded seats.
Candidate B1 received approvals from 1 voter, B2 received approvals from 1 voter, and B3 received approvals from 1 voter. None of the B candidates meet the approval threshold of 3 and are therefore not awarded any seats.
Therefore, candidates A1 and A2 are elected, while candidate B1 is not elected, fulfilling the requirement of electing two A candidates and one B candidate. This allocation is also proportional to the number of approvals received by each candidate.

OK, you've elected just two candidates again. I think you are just bad at this. We don't need to continue.

I apologize again for the mistakes in my previous responses. I strive to provide accurate and helpful information, and I will continue to improve my performance. If you have any further questions or concerns, please let me know.

@rob Well, I've not actually asked it to write code myself - it's more that I'm surprised it can given its failings in other areas. Interestingly I tested it out with some voting method stuff. I (optimisitically) wanted to see if it could come up with an algorithm that would produce a specific result for a specific set of ballots. I'm quite interested in approval PR methods so I started with that. But I started off with the most basic case. Just two factions with a 2:1 ratio of voters and 3 seats, and I wanted it to come up with a method that would give seats in a 2:1 ratio. If it had just said it couldn't do that, then that would have been fine. But it came up with a method, and it was just wrong. I pointed out the error, so it came out with another wrong method. I think I gave up at that point, because there was no way it was going to deal with the more complex criteria!

But there's other things as well. I'm quite into running and I was interested to see if it had anything to say about conversions for "equivalent" times for one distance to another. At one point it suggested that someone running 5km in 20 to 22 minutes could run a marathon in 2 hours 30. (2:30 for a marathon is the same pace as about 17:45 for 5km, so you'd have to actually be able to run 5km quite a bit faster than that.)

But this is just a small sample. It just seems to fall apart when there's numbers, so I would have thought it would struggle with the precision required for code.

@rob That's interesting, because I've generally found that ChatGPT is terrible at anything at all technical or requires precision. It's good at saying stuff. A BSer basically. That's why I'm always amazed when people say they've got it to write code.

@jack-waugh Is there any suggestion that this forum would not continue in its current form and that posts could be lost?

ChatGPT designed a restaurant for us.

Welcome to VoteVeg, the vegan restaurant that’s all about democracy and delicious plant-based food! Our restaurant is designed to promote different voting methods and encourage people to learn more about different systems of elections. Here’s a look at our menu:

Ranked Choice Salad
A refreshing blend of mixed greens, cherry tomatoes, cucumber, and red onion topped with your choice of protein (such as marinated tofu or seitan) and your preferred dressing. This dish is a great way to start your meal and get a taste for how ranked choice voting works.

Proportional Pizza
Our delicious pizzas come in many different flavors, including Margherita, Veggie Supreme, and BBQ Tofu. Each pizza is sliced into proportional sections to represent the number of seats a party or candidate would receive in a proportional representation system.

Approval Appetizers
Choose from a variety of tasty small plates, such as our roasted garlic hummus with fresh pita, crispy buffalo cauliflower bites, or vegan nachos. With approval voting, you can vote for as many dishes as you like, and the ones with the highest approval ratings will stay on the menu.

Plurality Pasta
Our classic spaghetti and meatballs dish has been given a vegan twist! This hearty plate of spaghetti is topped with vegan meatballs and a rich tomato sauce. With plurality voting, it's all about choosing your favorite option, just like choosing your favorite candidate.

Instant Runoff Ice Cream
Our dessert menu features a variety of flavors of vegan ice cream, including chocolate, vanilla, and strawberry. With instant runoff voting, you can rank your top choices, and we'll serve up the flavor with the most support!

Come dine with us at VoteVeg and learn more about the importance of fair and inclusive voting methods while enjoying delicious vegan cuisine.

@akazukin5151 First of all, I'd say that the Droop Quota isn't very good when the quota is so small. It's basically a rounding up of the Hagenbach-Bischoff Quota, which would be approx 3.23 in this case. So 4 is a massive difference from this (24% more I think). A more realistic quota for most elections would be in the thousands, so rounding up to the next integer would be a very small percentage difference. I think this is largely where the problem stems from.

But if we're going with 4 as the quota anyway, I think we're just minimising the amount over quota they go. So we have the following seats "owed" to each party:

A - 5.75
B - 6.5
C - 12.75

You'd end up with:

A - 7 (1.25 over)
B - 8 (1.5 over)
C - 14 (1.25 over)

This gives 29 seats and then there would be one last seat to go to either A or C (a tie) putting them 2.25 over quota.