https://forum.electionscience.org/t/sequential-multiwinner-voting-methods-visualized/773
If you haven't seen the previous ternary plot diagrams I have done for single winner voting methods and apportionment algorithms, you might want to check those out as well.
The fallowing ternary plot represents what portion each of 3 different groups of voters (the cyan group, the magenta group, and the yellow) makes up of the electorate:
The closer you are to one of the 3 corners the more the group of voters that corner represents make up of the electorate. If you are directly on one of the corners, then the entire electorate composes of just the group of voters that corner represents.
The three groups of voters represented in that ternary plot have the fallowing preferences:
| Cyan Candidate | Magenta Candidate | Yellow Candidate | |
|---|---|---|---|
| Cyan Group | 10/10 | 0/10 | 3/10 |
| Magenta Group | 7/10 | 10/10 | 0/10 |
| Yellow Group | 0/10 | 5/10 | 10/10 |
Like in my single winner voting methods visualized post and apportionment algorithms visualized post, each of the fallowing diagrams represents what the winners would be under different electorates. In each election the rules are modified such that each candidate can win multiple seats as if they were replaced by a clone everytime they won a seat. The methods I have chosen to compare are the two most established party agnostic proportional voting methods (STV and both versions of RRV) along with the 3 multi-winner method proposals the Equal Vote 0-5 PR Reaserch Committee I am on has came up (you can read up on each of those here: https://www.starvoting.us/star_pr) as well as one additional method: Sequential proportional score voting (which is just SPAV + KPT). Though I plan to create diagrams for many more sequential and optimal PR methods in the future.
Re-weighted Range Voting (D'Hondt/Jefferson version, 2 winners):
Re-weighted Range Voting (Sainte Lague/Webster version, 2 winners):
STV (2 winners):
SSS (2 winners):
SMV (2 winners):
Re-weighted STAR (2 winners):
Sequential Proportional Score Voting (2 winners):
Re-weighted Range Voting (D'Hondt/Jefferson version, 3 winners):
Re-weighted Range Voting (Sainte Lague/Webster version, 3 winners):
STV (3 winners):
SSS (3 winners):
SMV (3 winners):
Re-weighted STAR (3 winners):
Sequential Proportional Score Voting (3 winners):
Re-weighted Range Voting (D'Hondt/Jefferson version, 4 winners):
Re-weighted Range Voting (Sainte Lague/Webster version, 4 winners):
STV (4 winners):
SSS (4 winners):
SMV (4 winners):
Re-weighted STAR (4 winners):
Sequential Proportional Score Voting (4 winners):
Re-weighted Range Voting (D'Hondt/Jefferson version, 5 winners):
Re-weighted Range Voting (Sainte Lague/Webster version, 5 winners):
STV (5 winners):
SSS (5 winners):
SMV (5 winners):
Re-weighted STAR (5 winners):
Sequential Proportional Score Voting (5 winners):
And finally, this is what I speculate (and apparently what Toby Pereira also speculates) what the ideal proportions of the legislature C M and Y should control are if the legislature had a number of seats that approached infinity. This is also what I believe harmonic voting with the probabilistic transformation instead of the KP transformation reduces to in that case when there are infinitely many seats to fill (though the KP transformation and probabilistic transformation are equivalent when there are just 3 parties):
Though you can take that last diagram with a grain of salt, as it's just speculation. If you believe that the ideals of proportional representation should be strictly just to make the number of seats each party gets to be proportional to the 1st preferences of voters (like in a party list sense), then to you the ideal proportions would instead be equivalent to the 1st diagram (the one showing the different proportions of each of the 3 groups).