[EM] Summability and proportional methods

[Jameson Quinn]

Most proportional voting methods are not summable. Transfers, reweightings,
and otherwise; all of these tend to rely on following each ballot through
the process. This makes these methods scary for election administrators.

I know of 3 ways to get summability: partisan categorization, delegation,
and second moments. List-based methods (including partially list-based ones
like MMP) use partisan categorization. GOLD voting
<http://wiki.electorama.com/wiki/Geographic_Open_List/Delegated_(GOLD)_voting>
does
it by giving voters a choice between that and delegation. Asset voting and
variants use delegation.

The other way to do it is with second moments. For instance, if voters give
an approval ballot of all candidates, you can record those ballots in a
matrix, where cell i,j records how often candidates i and j are both
approved on the same ballot. This matrix keeps all the information about
the two-way correlations between candidates, but it loses most of the
information about three-way correlations. For instance, you can know that
candidates A, B, and C each got 10 votes, and that each pair of them was
combined on 5 ballots, but you don't know if that's 5 votes for each pair,
or 5 votes for the group and 5 for each. Note that those two possibilities
actually involve different numbers of total votes — 15 in the former, 20 in
the latter. In order to fix this, you can instead make separate matrices
depending on how many total approvals there are on each ballot — a "matrix"
for all the ballots approving 1, one for all those approving 2, etc. Thus,
in essence, you get a 3D matrix instead of a 2D one.

Once you have a matrix, you can essentially turn it back into a bunch of
ballots, and run whatever election method you prefer. The result will be
proportional insofar as the fake ballots correspond to the real ballots.
How much is that? Well, I can make some hand-wavy arguments The basic
insight of the Central Limit Theorem (CLT)  — that second moments tend to
dominate third moments as the number of items increases  — would seem to be
in our favor.

I think this could be an interesting avenue of inquiry. But on the other
hand, the math involved will immediately make 99% of people's eyes glaze
over.

If this is not possible, then the only 2 ways towards summability are
partisan categorization and delegation. GOLD uses both. For a nonpartisan
method, I don't think there's any way to be summable without forcing people
to delegate; and I think that forced delegation is going to be a
deal-breaker for some people.

So I'm frustrated in trying to design a nonpartisan proportional method
that's as practical as GOLD and 3-2-1 are for their respective use cases.
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