Should Bullet Votes be Spent First?

First, notice that this question only makes sense for methods with reweighting steps that incorporate quotas. Now, onto some potential merits and drawbacks.

Once a candidate is elected, bullet votes for that candidate have expressed no more interest in the outcome. If they are not used now, they never will be, and more unused votes suggests a less representative outcome. Also, other ballots that give points to that candidate still have expressed interest in the outcome. Allowing them to keep their weight increases their chance of being used to elect another candidate. This again suggests a more representative outcome.

Potential problems include the fact that adding bullet votes for a candidate whose election was already certain may then result in another candidate being elected even though they never intended for their ballot to benefit that candidate. On the other hand, they expressed no preference for the candidate who lost their seat either, so maybe this isn’t an issue. The other issue is that following this rule to the letter violates IIA, since a vote approving the elected candidate plus an irrelevant one would not be a bullet vote. It’s likely impossible for a sequential method to obey this rule without violating IIA, since it can’t tell which candidates are irrelevant until the election is over. Optimal Monroe does both, though.

If they are used, they have no weight to contribute to anyone’s election. If they aren’t used, the only candidate they supported has already been elected, so they are still contributing 0 weight to everyone else’s election. So either way, does it really matter whether they are used up or not? They were already included in the calculation of the Hare Quota, so passing or failing IIB is out of the question with this suggested modification. Also,

Not necessarily true. Take the example of

The first 3 winners are A, A, B in both SSS and Capped Quota SSS. The example is now (A 13 B 8 C 10 D 9). SSS elects A and C with deficits to give a winner set of (A, A, B, A, C), with a total of 83/100 points spent (A’s 53 + B’s 20 + C’s 10.)

Capped Quota SSS would see that A should win with a q-value of 17.666 (B’s q-value is 14, C’s is 10, D’s is 9), and then would elect B with a q-value of 14 (A’s is 13.25, C’s is 10, D’s is 9). So the winner set here is (A, A, B, A, B) with a total of 81/100 points spent (A’s 53 + B’s 28 = 81), 2 less than with plain SSS, but it’s a fairer outcome.

To give a much clearer, but farcical and Monroe-violating example of why spending more points doesn’t mean more representative outcomes, take the above example with SSS, but remove the surplus handling, and have the 9 D voters divide themselves into 5 D voters and 4 E voters. A would win with 53 points, B with 28, C with 10, D with 5, and E with 4, for a winner set of (A, B, C, D, E) with a total of 100/100 points spent, yet this is very unrepresentative compared to either (A, A, B, A, C) or (A, A, B, A, B).

Well, if you remove the surplus handling, it’s no longer a proportional method. Anyway, the cases I’m referring to don’t happen in cases where the voters and candidates can be grouped into parties. They look more like
1 quota A
1 quota AB

I may be getting this wrong, but your question seems to touch at least partially on this general difference (from this link https://www.reddit.com/r/RanktheVote/comments/cw7ep6/what_would_you_guys_think_of_sequential_monroe/eyd822m?utm_source=share&utm_medium=web2x):

To illustrate the difference between Delta based definition of support and an Absolute Score definition of support can be illustrated as follows:

Assuming that A is the candidate being seated. Which of the following is better served by being considered an “A” voter?

  1. A:4, B:5, C:3, D:5 (Delta: -0.25, Absolute: 4)
  • A:3, B:0, C:1, D:0 (Delta: 2, Absolute: 3)

To my thinking, voter 2 should be selected as part of A’s quota, because they like A by at least two points more than any other option. On the other hand, Voter 1 would be better served by either B or D…

Another example (from https://np.reddit.com/r/EndFPTP/comments/8nbqo0/counting_ballots_under_reweighted_range_voting/dzue0fz/):

My definition is as follows:

Score for Candidate X - Mean of Scores on that ballot 

homunq prefers the following (IIRC)

Score for Candidate

I will allow that his is simpler, but it rubs me slightly wrong, because I believe that someone who returns a ballot 5/4/3/4 (M: 1, H: 5) would be much better represented by B, D, or even C than someone who returned a 4/0/0/0 ballot (M: 3, H:4).

Thus to apportion the first ballot to A would do a greater disservice to the second voter (minimum loss of expected utility of 4) than apportioning the second ballot would do to the first ( maximum loss of expected utility of 3).

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