Another interesting property of the unfortunates class of method
What I mean by the "unfortunates class" is a Droop-proportional method
like this:
1. Use an appropriate method to elect n-1 of n
2. Eliminate the loser and repeat until n-1 is equal to the number of
seats you want.
I just realized that this class of method is house-monotone! Because
we're eliminating down to a given size set, the result for a smaller
election must be a subset of the result for a larger election.
In other words: nobody gets kicked out when going from a smaller seat
outcome to a larger seat outcome. So for all its flaws, this does show
that house monotonicity and Droop proportionality are compatible! I
wouldn't have thought that's possible.
There's a disadvantage, though. It means that every unfortunates-class
method has center squeeze in it, for the same reason that the Bucklin
method I described does. So these would all make pretty awful
single-winner methods.
And probably, by analogy to the difference between unconstrained and
hierarchical clustering, one can show that a good non-house monotone
method can produce much better results than the best house monotone method.
But still: I didn't know that was possible. I knew it was possible for
party list, but it's the kind of thing I thought couldn't possibly hold
for more complex ranked ballots.
On another note: I've been tinkering with Z3 and I think it says that
there exists no single vector v so that:
(fpB > 1/3 and fpC > 1/3 > 0) or
(lpA > 2/3) implies
v dot [A>B A>C B>C |V|] > 0
(i.e. "the loser according to v is A")
and
when rotating the candidates, there's never a situation where
the loser according to v is more than one candidate.
I.e. there's no way to, with a single vector, use pairwise counts alone
to create a method that ejects unfortunates and never says that two
candidates both lose.
Baby steps, but still... I suspect it's true in general: that pairwise
counts can't be used to discern such positionally based conditions.
-km
On 2/23/23 11:13, Kristofer Munsterhjelm wrote:
What I mean by the "unfortunates class" is a Droop-proportional method
like this:
1. Use an appropriate method to elect n-1 of n
2. Eliminate the loser and repeat until n-1 is equal to the number
of seats you want.
I just realized that this class of method is house-monotone! Because
we're eliminating down to a given size set, the result for a smaller
election must be a subset of the result for a larger election.
In other words: nobody gets kicked out when going from a smaller seat
outcome to a larger seat outcome. So for all its flaws, this does show
that house monotonicity and Droop proportionality are compatible! I
wouldn't have thought that's possible.
I probably should show that an unfortunates-class (Droop-proportional)
method actually exists, since I said I think the Bucklin-derived one is
only partly/pseudoproportional.
This is relatively simple. For step one, let "an appropriate method" be
STV. Since it passes the DPC, so should the unfortunates method based on it.
-km