So I came across this precinct-summable PR method that uses score ballots for v voters and c candidates. It's a bit complicated to describe, but here's my best attempt:

1. Each voter i casts a score ballot S_i.
2. Suppose S_i ranks n candidates top. Create a matrix M_i and vector w_i where M_i[a, :] = S_i / n and w_i[a] = 1/n if a is ranked top, else both are 0. (Effectively, we're combining all the votes that score each candidate top into one basket. If multiple candidates are scored top,
3. Let M be the sum over all voters i of M_i and w the sum of all w_i. (M needs to be normalized by w).
4. Let X be formed by dividing row j of M by w[j]. Run RRV (or its optimal form) where the rows of X are "ballots" and the elements of w are the initial ballot weights.

Example: The ballot (A=1, B=2/3, C=1/3, D=0) becomes

M_i =
  1 2/3 1/3   0
  0   0   0   0
  0   0   0   0
  0   0   0   0
w_i = [1 0 0 0]^T

while (A=1/2, B=1, C=0, D=1) becomes

M_i =
  0   0   0   0
1/4 1/2   0 1/2
  0   0   0   0
1/4 1/2   0 1/2
w_i = [0 1/2 0 1/2]^T

Of course, I don't like the discontinuity involved (in splitting between candidates that are scored "top") so I would change the calculation of M to be proportional to the elements of the voter's ballot (so that ends up looking like M_i[:, :] = S_i S_i^T and w = S_i/sum(S_i)). The ballot (A=1, B=2/3, C=1/3, D=0) then becomes

M_i =
   1  2/3  1/3   0
 2/3  4/9  2/9   0
 1/3  2/9  1/9   0
   0    0    0   0
w_i = [1/2 1/3 1/6 0]^T

I call this soft Simmons PR (by analogy to "softmax" which was partially an inspiration).

The nice thing is that since there are only c distinct "ballots" in the final step, there might be some kind of efficient special-case algorithm to determine the optimal-PAV (or its score counterpart) winner slate, although I am not completely sure of this.

(Can I not use LaTeX here?)

@andy-dienes said in An argument against committee monotonicity:

There are rules which are committee monotone which will elect AC (e.g. SAV)

That means that SAV elects C for k=1, which is definitely illogical. As for MES, part of that might be because it's a little unclear for which values of x the following generalization should elect AC instead of BC:

A (1-x)/2
AB x/2 - epsilon
BC x/2 + epsilon
C (1-x)/2

All-at-once d'Hondt PAV elects AC for x < 2/3. I think my argument is more persuasive if x is closer to (but still more than) 1/2, and epsilon is small compared to x-1/2.

@rob said in Terms for Specific Voting Systems:

considering that in French, they use the same word for both black and white.

Source? I remember black being noir(e) and white being blanc(he). Of course, it's possible there's some sort of slang term or something similar to our (in)flammable...

While we're on the topic of "terms for specific voting systems"... I think we need a better name for what's been called "Proportional Approval Voting". I feel like it should emphasize that the candidates are elected as a slate. "NP-complete PAV" seems like an attack. Perhaps "group PAV"?

I don't think this is original but it does not appear to be present on this forum yet.

Consider the following election:

A 20
AB 29
BC 31
C 20

If there is only one winner then clearly B is optimal. However, if there are two winners then {A,C} seems more logical than {B,C} since the former represents all voters while the latter only represents 80%, with 31% being doubly represented.

Optimal d'Hondt PAV gives {A,C} a score of 100 and {B,C} a score of 80 + 31/2 = 95.5, so {A,C} win. SeqPAV, of course, elects B first.

@rob said in Precinct summability of IRV:

Not sure I understand how that applies. The recount is not affected by the precinct submitting results as they come in. They still have all the ballots they can use for the recount.

Let's say there is a C1V, AV, and IRV election all of which are disputed.
The C1V election has a precinct that reported

A 571
B 482
C 6144
D 16
E 3
F 8

This can be easily checked by hand: sort all the ballots into piles for each candidate.

Now, consider an Approval Voting precinct:

G 883
H 476
I 340
J 181
K 1105

To verify this by hand, you'd need to check all ballots 5 times, since any individual ballot could approve multiple candidates.

Finally, consider an IRV precinct:

L,M,N,O 366
L,M,O,N 68
L,N,M,O 15
L,N,O,M 70
L,O,M,N 4
L,O,N,M 53
etc.

That would involve sorting the ballots into 24 piles to check the totals, or 4 passes through the ballots to check the winner, but if the latter check fails you have a crisis. (For example, suppose the "election night" results are L=50k, M=76k, N=60k, O=L+6. L is eliminated which ends up giving, say, O the victory after N is eliminated. But a recount finds an extra dozen votes for L, so O should have been eliminated first!)

(Actually... verifying the totals may not be as hard as I think. After the L-top ballots are separated, you can verify they add to 366+68+15+70+4+53. Then within the L-top ballots you can verify that 366+68 rank M second, 15+70 rank N second, and 4+53 rank O second.)

I think the argument is that if you had 20 candidates, then a malicious agent "Mallory" could tell "Alice" the following:

I want you to vote [Adolf Hitler, Josef Stalin, Benito Mussolini, Joe Smith, John Doe, Jane Brown, Jack Random, Jeff Specific, Jen Order, Jake Candidates, Winston Churchill, Franklin Roosevelt]. If I see that precise order in your precinct, then I will {...}, else I will {...}.

It's also been argued that this is difficult to verify by hand in the case of a recount. However, if you're just trying to verify that the IRV (or seqPAV or RRV) winner is correct, you can do it in (# rounds) serial counts (meaning each precinct has to do a separate count for each round, but they can be done in any order) if the supposed order of elimination is known. (If there happens to be a disparity then you have a problem, but that's Somebody Else's Problem.)

This is a topic I want to follow because it will make or break the... morality of my much-desired multiwinner Approval research.

Greetings citizens of Earth. We come in peace.

I am a rogue mathematician trying to hold on to moral courage in a strange world. I'm interested in researching proportional voting methods. Where can I help?

(OK, I admit I'm a little bit spooked by @Sass' impassioned calls for precinct summability in high-profile elections. But there might still be hope?)