Interesting example where Apportioned Score Voting fails one interpretation of PR

Number Ballots
34 A1:5 A2:5 A3:5 B1:3 B2:3 B3:3 C1:0 C2:0 C3:0
15 A1:2 A2:2 A3:2 B1:5 B2:5 B3:5 C1:2 C2:2 C3:2
51 A1:0 A2:0 A3:0 B1:2 B2:2 B3:2 C1:5 C2:5 C3:5

Hare Quota is 33.333 voters, and 166.666 points. The A-faction has this, but the final result is actually 2 C 1 B.

With Sequentially Spent Score with Monroe selection (elect whoever can get a Hare Quota of points with the fewest ballots + other tiebreaker, and I think you have to reweight only the ballots used to select the winner), the A-faction would for sure be able to win a seat. As a bonus, this is not too complex to explain either.

I could be wrong about how ASV works, so I’ll update if that Reddit thread proves the example wrong. For example, it may be possible to get an A winning with the “Score - Mean of Scores on Ballot” metric for apportionment.

Which PR criterion do you think it violates. The standard party PR criterion only counts ballots as a party when they give extreme “binary vote” scores.

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Didn’t know that, sorry.

So an interesting modification has been proposed to fix ASV here (https://www.reddit.com/r/EndFPTP/comments/d5t10w/i_submitted_a_california_ballot_measure_to/f0wqqp4?utm_source=share&utm_medium=web2x):

That depends on whether “Difference from Average” considers those candidates that have been seated, and your example is a compelling argument that it shouldn’t.

In block 2, you’d have the following got the following:

Number|Ballots|With Seated|Diff for B|W/O Seated|Diff for B| :—|:—|:-:|:-:|::|:: 34|A1:5 A2:5 A3:5 B1:3 B2:3 B3:3 C2:0 C3:0|(15+9)/9==2.(6)|3-2.(6)==0.(3)|(15+9)/8==3|3-3==0|A1:2 A2:2 A3:2 B1:5 B2:5 B3:5 C2:2 C3:2|(6+15+6)/9==3|5-3==2|(6+15+4)/8==3.(3)|5-3.(3)==1.(6) 17.66…|A1:0 A2:0 A3:0 B1:2 B2:2 B3:2 C2:5 C3:5|(0+6+15)/9 == 2.(3)|2-2.(3)==-0.(3)|(0+6+10)/8==2|2-2==0

In both cases, you’d take from the B voters first, and if we consider the Seated candidates in the Delta, the the 8.(3)% required fill out the quota from A, as in your calculations.

If we don’t consider the seated candidates, the remainder of the quota comes from the other two factions proportionally, call it 2:1, for simplicity’s sake (erring to the detriment of the A faction), and you take 5.41(6) from A, and 2.91(6) from C, leaving:

Number Ballots
28.58(3) A1:5 A2:5 A3:5 B2:3 B3:3 C2:0 C3:0
14.75 A1:0 A2:0 A3:0 B2:2 B3:2 C2:5 C3:5

Obviously, then, A would win, with the final results being [C,B,A]. C and A both have one full Hare Quota of unique top preferences, and, and they both have one seat each, satisfying the assertion.

Another thing about this example is that it’s not at all clear that ABC is the best outcome. Not only is BBC a credible alternative, ACC is as well- in fact the “Droop Proportionality” standard that ranked methods are held to would require ACC.

Getting ABC in this scenario is actually quite difficult- the type of method that gives A a seat (Monroe-like methods) will likely give C two seats. The type of method that gives B a seat (“utilitarian selection”) likely shuts out A. To see why, consider what happens if A wins- who pays for it? Mostly A votes, maybe a few B votes (A votes contributed to 85% of A’s sum score), but certainly no C votes. A barely made quota, so the A ballots are going to have to pay until they are insignificant. This will not only end any chance of A getting a second seat, but harm B as well- A voters preferred B to C as strongly as B voters did (also as strongly as C voters prefer C to B.) Assuming vote unitarity, there would be 15 1/3 votes preferring B to C by 3 points, less than triple the number of C votes. So C probably gets both seats.
In contrast, if B wins a seat, the A voters most definitely have to pay for some of it, since their secondary support for B is an important part of B’s strength (they contribute more total points to B than B voters themselves.) This takes A below a quota, and since the strength of A votes’ preference of A over B was weak to begin with (13.6 votes worth), it likely means that A won’t have the power to push A over B after the inevitable C seat.

I think BBC works, but in practice I’d expect most fans of proportionality would prefer an AC? outcome over one without A.