Independence of Universally Approved Candidates v Top Tier Proportional Approval Methods
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Under Independence of Universally Approved Candidates (IUAC) (see also Universally Liked Candidate Criterion, Strong PR), adding in some universally approved candidates and extra seats to accommodate them should not change the outcome for the other candidates. E.g. with 6 to elect:
2: U1-U3, A1-A3
1: U1-U3, B1-B3The result should be U1-U3, A1-A2, B1. However, Thiele's PAV elects U1-U3, A1-A3.
Fully optimal methods like Optimal PAV and COWPEA (which allow candidates to be elected in any proportion) completely bypass the IUAC criterion by electing only the universally approved candidates. The lottery versions of these methods also pass by electing the universally approved candidates to the first positions and then electing proportionally from there.
However, one "normal" method (fixed number of candidates elected with equal weight) that does pass is the specific version of var-Phragmén where a candidate's load must be spread equally across its voters. I've always called this Ebert's Method, but I'm not sure he actually intended it to mean this. Anyway, this method is poor overall as it is non-monotonic and can give disproportional results, but I think it does give good results with certain toy examples such as the one above and can be enlightening.
Anyway, while COWPEA and Optimal PAV don't have to deal with IUAC, I did wonder what might happen if a certain group of voters did all approve a particular candidate but not the entire electorate. Would these methods pass this lesser form of IUAC? E.g.
1: U
2: A
1: B
4: UA
2: UBThe extra single U voter at the top is to stop a method from only electing A and B, and forcing it to interact with all 3 candidates. So should the A:B ratio be 2:1 here? Why? Why not? What do each of our methods do? First of all PAV's ratios:
A: 0.376697
B: 0.147247
U: 0.476056This gives an A:B ratio of 2.558:1
COWPEA:
A: 27/75 (0.36)
B: 10/75 (0.133)
U: 38/75 (0.507)This gives an A:B ratio of 27:10 or 2.7:1. This is even harsher than Optimal PAV.
Finally Ebert's Method:
A: 1/2
B: 1/4
U: 1/4This gives an A:B ratio of 2:1, which is what might seem intuitively correct.
So are Optimal PAV and COWPEA "wrong" here? My thinking of late has been that these two are the ultimate in optimal methods (while differing slightly philosophically with each other) and very trustworthy, so I'd be surprised if these results are indeed "wrong" in some way. So why these results?
I wonder if it's to do with monotonicity. The result from Ebert's Method gives U a very low ratio compared to the others despite it being the most approved candidate. Obviously it would be possible to make this higher while keeping the A:B ratio the same, but there might be other reasons not to do this. Calculating the total approval rating of each slate of candidates under each result, you get:
PAV: 6.034
COWPEA: 6.107
Ebert: 5.5So the total score from the Ebert result is just 5.5, quite a bit lower than the others with COWPEA coming out on top. I think this has been quite an interesting experiment.
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@toby-pereira this seems like a tricky thing to deal with. I appreciate your posts here, PR methods seem to be pretty complex. The whole process seems to be a “cake cutting” problem, and a while back, I was trying to think about how to pose the situation in a way that enables the optimal cake cutting protocol to work. It might not be clear, fully hashed out or easy to follow, but I wonder what your thoughts about the concept are: