# 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 Iâ€™ll update if that Reddit thread proves the example wrong. For example, it may be possible to get an A winning with the â€śScore - Mean of Scores on Ballotâ€ť 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 â€śbinary voteâ€ť scores.

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Didnâ€™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 â€śDifference from Averageâ€ť considers those candidates that have been seated, and your example is a compelling argument that it shouldnâ€™t.

In block 2, youâ€™d have the following got the following:

Number|Ballots|With Seated|Diff for B|W/O Seated|Diff for B| :â€”|:â€”|:-:|:-:|::|:: 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) 17.66â€¦|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, youâ€™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 donâ€™t consider the seated candidates, the remainder of the quota comes from the other two factions proportionally, call it 2:1, for simplicityâ€™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 itâ€™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 â€śDroop Proportionalityâ€ť 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 (â€śutilitarian selectionâ€ť) 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 Aâ€™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 Bâ€™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 votesâ€™ preference of A over B was weak to begin with (13.6 votes worth), it likely means that A wonâ€™t have the power to push A over B after the inevitable C seat.

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