Possibly, a sub-election could be run to determine the representatives within the A:45 group, etc. Maybe they could be given 4 seats, the B:35 group 3 seats, and C:20 2 seats. That could refine representation, some candidates might pick up multiple seats. It’s probably getting messy and complicated, it essentially becomes a hierarchical partitioning of ballots. I’m not sure what to make of that prospect, it starts looking like a network/phylogenetic tree or forest architecture, and that can become arbitrary fast.

I do see what you mean. An individual voter may actually prefer a particular coalition of candidates, rather than just want to get their top guy in. I wonder if non-strict rankings and distributed power would mitigate this issue, or for instance, if the A:45 group's sub-election guaranteed a seat for A, and ran the election on the remaining candidates, that might align with the spirit of preference for whole coalitions.

You definitely are more familiar with this space than I am, so I wager some of my objections may be non-issues when one considers alternative PR methods. But it seems to me that in this case, pushing for coalitions that respect single individual preferences for whole coalitions can lean toward reduced diversity and reduced minority representation. Is that inaccurate? Or is that a common tradeoff issue in PR systems?

“Would the weighting purely count towards their voting power in the elected body, or does it have other effects such as more time to speak?”

Yeah, it does beg some questions.

EDIT: After multiple adjustments made to guard against clone dependence and tactical voting, there is a non-monotonicity issue in my latest version (/branch, it is not my original concept so I don’t claim ownership in any way), where a minority faction can gain strictly preferred representation in the form of a seated candidate by merely withholding approval for that candidate. I have a concrete example of this, and may try to see how to address it. It may be due to something unnecessary.

Ranked voting - Wikipedia

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

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

Condorcet methods: Wikipedia, Carneades.org video

Borda Count: Wikipedia, Carneades.org video

Majority criterion: Wikipedia, Becky Moening video

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

Condorcet winner criterion: Wikipedia, Carneades.org video

Monotonicity criterion: Electowiki article, Becky Moening video

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

Manipulability: Wikipedia article on strategic voting, Katherine Heller video

Random dictator: Wikipedia article, Carneades.org video

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.

Under the lazy fingers assumption, in IRV, any voter whose favorite candidate is one of the 2 finalists (and the finalists were foreseeable) will bullet-vote.

The opposite assumption is the "expressive voter assumption", which assumes that voters will add rankings despite knowing that it has a 0% chance of affecting the election.

@lime trying to imagine what PB-VCG would look like practically in this context, given we're electing voters to offices of varying power rather than fulfilling projects with known costs. Each member votes with virtual assets, single offices "priced" and allocated with normal Vickery, set of comparable offices with VCG?

Yes, although not 100% sure what you mean by virtual. If every MP had $1 million in budget to spend (e.g. because we're using something like MES), then you can bid with that budget. Payment involves transferring part of your share to another MP.

Auction order would have a large effect.

I was thinking of a single auction to elect the whole cabinet simultaneously.

Would expect many offices to have multiple all-in bids, which is problematic when they're using equal virtual assets.

Because of collusion, you mean?

According to my wikipedia education, one of the issues with SPAV compared to PAV is that it is a 'greedy algorithm' that only looks at the most locally optimal solution.
Now SAV considers the whole field at once but doesn't fix under and over votes.

AFAIK Sequential SAV and Sequential PAV are the same procedure, which is actually the trick I use to explain SPAV more intuitively: I describe SAV, then explaining how SPAV fixes the spoiler effect in SAV.

Did you read the SAV Wiki article in the past week or two? I tried explaining this way of thinking about SAV in the article to make it simpler, but might have unintentionally made it more confusing instead of less.

In an election with only two candidates, the Condorcet loser criterion implies the majority criterion. Given a three-candidate Condorcet cycle, it's always possible to eliminate a candidate who didn't win so that the winner changes. Thus the Condorcet loser criterion is incompatible with independence of irrelevant alternatives.

Say you've got an A>B>C>A cycle. Without loss of generality, A wins in some method that passes the Condorcet loser criterion.

Now remove B. C beats A head-to-head so must now win (A is the Condorcet loser with only A and C). Given that removing B has changed the winner from A to C, the method must fail IIA.

I figured as much in regards to reversing the numerical order from rating to ranking. My rationale for doing so was just to eliminated the strategic element of voters not using the entire range of scoring that a traditional scoring method provides. Although I suppose the same rule could be applied to the scoring method provided the number of ratings reflected the number of candidates rather than being a set (0-5), (0-9), or (0-99), scale. This seems like it would be a good way to circumvent min-maxing candidates without having to incorporate a top two runoff as in STAR, or having to use binary ratings as in Approval. I figured it probably hasn’t been popularized for a reason though, so thank you for mentioning borda counts so I can do some further research into it 🙂

@isocratia I’m not sure how score and approval can be said to solve the problem of burial at all, if one considers bullet voting to be a form of it.

I think burying is generally seen as putting a candidate underneath other preferred candidates. Normally in score/approval they'd be equally scored zero or unapproved.

If 'None of the Below' wasn't able to stop the count then it could only block one seat from being filled. I agree that's a simpler and more transparent implementation, but it seems like the intention is to be able to block multiple candidates if needed.

I think I mentioned in my post above that you could have "none of the below" (NOTB) standing effectively as multiple clone candidates. For example, it's a 5 seat election using score ballots. Whatever score someone gives to NOTB counts towards 5 clone candidates. So it's effectively a party vote for the NOTB party.

Or if it's ranked ballots, it would just take up 5 spaces in your rankings. So if your rank was 1. NOTB; 2. John Smith, it would actually be 1-5. NOTB; 6. John Smith.

I'd say that the Droop Quota isn't very good when the quota is so small.

I agree, I'm not allocating seats in a real life election so this is just purely theoretical.

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.

So that's the core reason, makes sense to me.

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:

That's pretty smart. This is the algorithm as I understand it:

If there are still seats remaining to elect, award seats to the party with the smallest difference between seats won and "party quota" (seats_won - votes / quota). Repeat until all seats filled

(For simplicity I'm breaking ties in favour of the first party)

So my example would start from the automatic seats [5, 6, 12]. Award seats based on the largest remainders method as usual, stopping when all parties has been awarded a seat: [6, 6, 12], [6, 6, 13], [6, 7, 13].

The remaining 4 seats would be filled like this

[7, 7, 13], [7, 7, 14], [7, 8, 14], [8, 8, 14]

Thank you very much for the help!

I know there are many criteria for voting methods (such as Majority, Condorcet, etc.) but nobody seems to have described a lot of criteria for vote-counting methods; precinct summability seems to be the only that I can think of off the top of my head. There are some criteria which I have thought about specifically in the context of pairwise counting which I would like to share. If anyone has any resources or ideas regarding vote-counting method criteria, I would be interested to learn more about them as well.

=Pairwise counting=

One of the strange things that I noticed about Condorcet methods is that their vote-counting is unbearably tedious even in scenarios where intuition suggests it ought to be fairly easy to count the votes. Consider the (completely theoretical) example of a voter who bullet votes in a Condorcet election where there are, say, 5 million candidates running. We would hope/expect that a vote-counter processing this voter's ballot would not require a lot of time to do so; however, the vote-counter would have to process almost 5 million pairwise matchups in order to finish counting the voter's ballot to then be in a position to discard the ballot.

• One could say that there ought to be an Independence of unranked candidates criterion stating that a vote-counter should not have to put any additional work in to process a voter's ballot because of "also-ran" candidates the voter didn't rank on their ballot; traditional pairwise counting obviously fails this criterion.

Based off of this, there are a few additional criteria I would propose for vote-counting methods, such as a "Bullet Voting" criterion: when a voter bullet votes (provides full support to 1 candidate and provides no information on their preferences for the other candidates), then the vote-counter should be able to process that voter's ballot using only 1 'mark'. (I will use the term 'mark' to indicate a tally mark, or a vote-counting operation done by the vote-counter). Among pairwise counting methods, the traditional pairwise counting method fails this criterion (in the above example, the vote-counter would need to make 4,999,999 tally marks to process the bullet-voter's ballot), but the "negative pairwise counting" method which I invented, as well as its variants such as "semi-negative pairwise counting", satisfy this.

• Further criteria can be invented like this: suppose we generalize this Bullet voting criterion to be an Approval voting-style ballot criterion: "if a voter votes in such a way that their ballot can be turned into a standard Approval voting ballot (that is, they give full support to some candidates and no support to all others), then the vote-counter should only need to tally 1 mark for every candidate the voter gave full support to, and be able to tally 0 marks for the other candidates."
  • And then we can further generalize this criterion in the context of score ballots to account for the possibility of voters giving less-than-full support to the candidates that they "approve" or more-than-zero support to the candidates they "disapprove" (for example, there is the theoretical scenario where there are only 2 candidates in the election and the majority chooses to compromise by not making a full distinction/strategic vote between the two candidates.)

During the invention of negative pairwise counting, another criterion popped out to me because of something I noticed: in a "pure" implementation of negative pairwise counting, a vote-counter would have to make tally marks for a candidate that a voter had explicitly given no support to on their ballot (i.e. they had ranked the candidate last, but in an explicit way, as opposed to not ranking the candidate at all and thus implicitly ranking the candidate last). This is because such a candidate would be explicitly ranked below all others, and thus 1 tally mark would have to be made for every candidate ranked above the last-ranked candidate, as well as 1 tally mark for the last-ranked candidate themselves. So this leads to two more criteria, which one could consider to be almost like the inverse of the Bullet voting and Approval voting-style ballot criteria: a Solo-no-support criterion stating that a candidate explicitly ranked below all other candidates should not require any tally marks to be made to account for them, as well as a more generalized form of this criterion to deal with multiple candidates who are explicitly given no support.

• To satisfy these two new criteria, I had to come up with a modified form of negative pairwise counting in which last-rank candidates are explicitly ignored by vote-counters (though that in and of itself is an undesirable trait in a vote-counting method, since the vote-counter would have to check which candidates are last-ranked before or while processing a ballot, and thus we could say that the modified negative pairwise counting method fails the imaginary criterion stating that a vote-counter should not have to explicitly know which candidates are last-ranked or not when processing a voter's ballot).

=Miscellaneous=

Other criteria or things for which criteria ought to be invented, or which could maybe be expanded upon:

• When dealing with write-in candidates, things can get complicated (at least for pairwise counting methods). So some criteria could probably be invented regulating how easy it ought to be for a vote-counting method to handle write-ins.
• There probably already are criteria or math describing how much data needs to be recorded at each step of vote-counting, but it might be interesting to see more study on this.
• The vote-counting methods for PR methods probably have some fascinating properties of their own that could be studied.
  • I actually had described in the old voting-theory forum an alternative way to do the vote-counting for certain sequential cardinal PR methods. For example, with SPAV, the normal way of vote-counting is to do several rounds of vote-counting, with the data taken from voters' ballots in one round completely ignored in future rounds; I envisioned an alternative vote-counting method (partially inspired by my negative pairwise counting idea) in which, for the first round of vote-counting (i.e. the round of vote-counting used to decide the first winner), the vote-counting is done as usual, but then in successive rounds, we simply subtract from the support recorded on a voter's ballot for each of their approved candidates as that voter begins to see some of their approved candidates get elected, rather than having to entirely re-count all ballots.
    • An example of this would be if you had 30 voters in SPAV who approved AB and disapproved CD, while the other 70 voters in the election all have different preferences and had all disapproved 'candidate A'; supposing that in the first round, A is elected, you would now reweight the 30 AB ballots to 1/2 weight (i.e. they are now treated as having the power of only 15 ballots), subtracting 15 votes from the vote tally for candidate B (since the 30 ballots that had supported B are now treated as having half-weight, or having lost the equivalent of 15 ballots' worth of weight), while not having to count any other ballots or change any other aspects of the tally.
    • The advantage of this approach is that only those ballots which have had one of their candidates elected in the immediately previous round need be (re-)counted in the current round, rather than having to re-count all the ballots.

floor( vote / droop_quota ) is the lower quota, ceil( votes / droop_quota ) is the upper quota.

So what is that algorithm? I mean, it could be Condorcet, and I have no problem with that, but I can't see how the term "proportional representation" applies. It just sounds like multi-winner.... there isn't anything proportional about it.
The only way the term "proportional representation" would apply (by my understanding of the term) is if we assume that there are some number of parties, and each candidate and each voter is in one and only one party. If a 3rd of the voters are in the Bull Moose party, then a third of those elected should be in the Bull Moose party. The further you get away from that, the less "proportional representation" seems to be a meaningful descriptor.
All of my complaints regarding PR (and with so many people's insistence that it is so much better than single winner methods such as Condorcet methods) are based on the assumption that voter X below is considered to have "better representation" if d, f and e are elected (because candidate d is very close to voter X), than if a, b and c are elected.

For the algorithm, you will be aware of Single Transferable Vote, which gives PR without any mention of parties. If voters happen to vote along party lines, it gives party PR, of course. That's just one example, without having to mention obscure methods invented by people on this forum.

And as for a, b, c versus d, e, f, I discussed that in the other thread here. I'm not sure it's worth quoting though because it's quite long.

If the ballot set consists only of

C1>C2>…>CN
CN>C(N-1)>…>C1

which are perfect reversals of each other, then a system satisfies the compromise criterion if and only if it elects either C((N+1)/2) if N is odd, or one of C(N/2) or C(N/2+1) if N is even.

For example, IRV does not satisfy the compromise criterion: if the ballots are

x>y>z
z>y>x

then the compromise candidate y is immediately eliminated.

Bucklin voting does satisfy the compromise criterion. Borda count does not, but concave score systems do. Plurality fails. Generally Condorcet methods are agnostic on the compromise criterion, and whether they satisfy it or not depends on how situations without Condorcet winners are handled. For example, Black's system (Condorcet//Borda) fails. Approval voting “conditionally" passes or fails depending on “approval thresholds” (technically the criterion does not apply, since it is defined for rank order systems).

For three candidates we can generalize and strengthen the compromise criterion:

For every ballot set of that is a multiple of

xyz+Pyzx+(1-P)zyx

for any P in [0,1], the winner is y. In this case, a weak Condorcet method that prioritizes candidates with positive margins of victory over ties and that satisfies the weak compromise criterion will also satisfy this stronger compromise criterion.

@rob I revisited this and I think Delegated Condorcet is a really interesting idea, what are your latest thoughts about that concept?

I haven't given it a lot of thought in the years since. It seemed like a good idea, and probably would work pretty well if adopted, but I doubt people would trust it enough to adopt it in the first place.

It might be best for local elections where there isn't a huge amount of publicity and the public tends to only know one or two candidates out of a larger field.

Agreed that voters' ballots will likely change depending on the candidates in the race in Approval more with a higher probability than in most other voting rules.

@Toby-Pereira to answer your question I'm looking into it. Arrow's Theorem actually has quite a few (slightly different) formalizations, and it looks like what I said is technically not true for the version defined on Wikipedia since that one only allows (strict) linear orders, but I feel quite sure I saw a formalization where the domain was all (weak) linear orders. I will try to find it.

In principle, this is an infinite recursion.

And hence, "hall of mirrors."

In practice, my iterative vote simulator stops when the ordering of candidates doesn't change between two rounds. Since the ordering is all the "vote caster" algorithm looks at, there is no point continuing because it will always get the same result.

But keep in mind, that doesn't mean it has reached the one and only equilibrium. There can be multiple equilibria.

But again, I'd rather have a method that doesn't encourage this sort of thing. If I did a vote simulator for Condorcet, there would be very little, if any, iterating, since there isn't an obvious way to adjust your ballot even if you know how others will vote.

[edit]

Suppose we have the system I described above, with the rule of matching ballots and throwing them out in pairs.

Let's say the candidates are Gore, Bush, and Nader and I vote Nader, 511. Let's account for the votes in terms of their effects on the pairs of candidates. In fact, let's ignore Nader-Bush and concentrate on Nader-Gore. What is the effect of my vote on the Nader-Gore accumulator? It has to be canceled by a vote of Nader, -511.