[EM] Summable Reweighted Bucklin Voting

[Jameson Quinn]

I suppose I wasn't clear enough about SRBV.

Here's the full procedure, with some quick justifications in parentheses.

1. Voters vote all candidates as "Preferred", "Approved", or
unvoted/disapproved. Write-ins are fine.

2. From just the preferred markings, you construct a 3D matrix where cell
x,y,z shows how many voters approved both x and y out of a total of z
approvals. That is, "Layer z" of the 3D matrix is a two-way candidate
correlation matrix (or a Condorcet matrix - it carries the same info in
different form) for the set of ballots with exactly n approvals.

(Layers 1 and 2 have full info, for the others there could be various
combinations of ballots which would lead to the same matrix. For instance,
in layer 3, with six candidates A-F, the ballots ABC, CDE, EFA, BDF give the
same matrix as BCD,DEF,FAB,ACE.)

3. If any of sum(Z|x,x,z) (that is, the approval score for candidate X) is
greater than one droop quota (call it d), elect and discount.

4. To discount layer z, first subtract d * (x,x,z) / sum(Z|x,x,Z). Then
subtract proportionally the same fraction from (x,y,z) and (y,x,z) with
x!=y. Then take the same actual amount that you subtracted from (x,y,z) and
subtract it from (y,y,z) - call that amount s(y,z). Now, for every pair m !=
n [ != x ], subtract from (m,n,z) the amount (m,n,z) * AVERAGE((m,m,z) /
s(m,z), (n,n,z) / s(n,z)).

(This is the "highest entropy" subtraction that preserves the matrix
invariants - that is, that sum((x,x)) * (z-1) = sum((x,y) for y!=x). It is
not the only subtraction that preserves these invariants - for instance, one
could have randomly picked ballots to subtract out. This also means that
SRBV is not quite identical to RBV. However, as the high-entropy solution,
the SRBV subtraction is the "most probable SRV" result over all possible
ballot combinations which give the same matrix (as you go to the limit of
infinite voters in the same proportions). I think, though I haven't yet
proven, that SRBV is proportional in all cases; certainly, I can't construct
an example where it doesn't, though I've tried. RBV itself is definitely
proportional.)

5. Repeat steps 3 and 4 as long as possible.

6. If there are still empty seats, repeat step 2 (construct another matrix)
with the approvals and preferences lumped together.

7. Repeat step 4 for each candidate already elected, in order.

8. Repeat steps 3 and 4 as long as possible.

9. If there are still empty seats, repeat steps 3 and 4 until you fill them,
except just choose the highest approval in step 3 and then in step 4,
instead of discounting by a Droop quota (and thus including negative
numbers), just zero out the approval for the elected candidate.

...

If you want to avoid {or minimize} electing any candidates with less than a
Droop quota, as in step 9, you can require voters to approve at least half
of the candidates plus half of the number of seats {or a lesser minimum
number, if you just want to minimize step 9}. To make that easy, you can
include some means on the ballot to simply approve certain predeclared party
slates at once. If this rule leads to certain "undervoted" ballots, you can
still count these ballots with the undervoted approvals distributed evenly
(fractionally) among the remaining candidates - that weakens the votes, but
does not discard them entirely, and they still count fully in round 1.

This method is monotonic, proportional (I think strictly, certainly as a
strong tendency), simple to vote, and summable. In the single-winner case,
it simply reduces to two-rank Bucklin - an excellent system. Actually, you
could easily generalize the above to 3-rank SRBV, but I think that's missing
the point of Bucklin. Two ranks is just the right expressivity to make
honest voting easy and (nearly) optimal. With less (approval), you're
literally forced to strategize your cutoff; with more (3 or more rank
Bucklin), you have too many confusing and potentially strategic choices.

JQ
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