Here's a short post (since I don't have as much time as I would like)
with an idea of how to make Sainte-Lague even more like STV. I started
thinking about it as part of my thinking that "perhaps pairwise
multiwinner methods will always be too complex"; and so I tried to
include some Condorcet compliance here as well.
Start with the original Sainte-Lague count. For each party, call the
difference between how many voters voted for them, and how many voters
they'd need to get the number of seats they currently did, the excess.
Then, as in STV, start redistributing the excess. Move a few voters at a
time (I don't know how many you can safely move in a batch) from their
first preference to their second.
(Note that for parties that got no seats and can get no seats through
redistribution, this has the same effect as elimination: their seat
count is 0 and so they can get that number of seats with no voters at all.)
Now the question is in what order to redistribute. I can think of three
ways. The first is in reverse Condorcet order: you redistribute the
voters for Condorcet losers first. The second is from parties with few
seats; and the third is whoever has the greatest excess at any point.
The method stops when no more redistribution can be done. This is also a
vague idea, but I guess something like "maximal excess is minimal" could
work... though I then would have to use more rigorous mathematics to
show that the method actually optimizes that.
The point of doing it in reverse Condorcet order would be to reduce to
Condorcet in the single seat case. Consider an L>C>R situation with
Condorcet social ordering C>L>R and where L gets the initial seat. Then
all the R-votes are redistributed to C since R can't win anyway, so L
loses its seat. At this point no further redistribution in this
direction can alter anything (we can only distribute from L to C, not
vice versa), so we'd like to finish there.
The few-seats order of doing it has an intuitive IRVish appeal: small
parties are disqualified/redistributed first, then larger ones.
Finally, the "greatest excess at any point" may have some desirabla
steady-state properties, and may get closer to optimizing "maximin
excess". I am not sure of this, though - it just sounds like something
that would. I also think that one would elect C in the example above: R
would have greatest excess (all the R-voters since they didn't get
anything). Enough R-voters are distributed to C to make C win. Then L
has the greatest excess and is redistributed to C as well, and it ends
when they all have equal excess.
Could this idea be developed into a method that would be better than
ordinary Sainte-Lague, yet also not as complex as my pairwise methods?
Perhaps. But I have little time and so don't know yet. I thought I would
just let you all know of the idea!
And it probably would not be cloneproof, (weakly) monotone or summable.
But I don't know of any method that follows STV's algorithmic template
that is.
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