Kristofer Munsterhjelm wrote:
I think that one problem with devising a multiwinner method is that we
don't quite know what it should do. PAV type optimization methods try
to fix this, but my simulations don't give them very favorable scores.
If we are to construct a multiwinner method that degrades gracefully,
we probably need to have an idea of what, exactly, it should do,
beyond just satisfying Droop proportionality (for instance). The
problem with building a method primarily to satisfy a certain
criterion is that if the criterion is broken slightly, then the
criterion does not tell us how the method should work; and therefore,
we might get "discontinuous" methods where the method elects a certain
set if a Droop quota supports it, but a completely different group if
the Droop quota less one supports that set.
So let's consider a case one may use to justify Condorcet, or to
classify Condorcetian methods: if politics is one dimensional, and
people prefer candidates closer to them on the line, then there will
be a Condorcet winner, and the CW is the candidate closest to the
median voter, and (if we think electing the CW is a good idea), we
should elect the candidate closest to the median voter.
This case, or general heuristic, seems to be simple to generalize, and
one may do so in this manner: call a position the nth k-ile position
if n/k of the voters are closer to 0. Then, a multiwinner method that
elects k winners should, if politics is one-dimensional, pick the
candidate closest to the first (k+1)-ile position, then closest to the
second (k+1)-ile position (first candidate notwithstanding as he's
already elected), etc, up to k.
To be more concrete, in a 2-candidate election, the first candidate
should be the one closest to the point where 33% of the voters are
below (closer to zero than) this candidate, and the second candidate
should be the one closest to the point where 67% of the voters are
below this candidate, the first candidate notwithstanding.
This is exactly what STV-CLE does.
However, I would choose a goal of minimizing the "mean minimum political
distance" (expected average distance between a voter and the nearest
winning candidate). On a uniform linear spectrum, the minimum is 1/(4k),
which occurs by electing the set {(n-1/2)/k for n=1 to k}. This reduces
to {1/2} when k=1, so is a generalization of the Condorcet Criterion.
However, it can also be applied to a multidimensional political spectrum.
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