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 heuristic covers only the one-dimensional case, but it is at least
continuous if the comparison of a particular election *is*
one-dimensional, and thus should reduce discontinuity problems.
Plurality party list methods can be modeled as instances where each
voter is located at the position of the party they voted for, since that
is all a Plurality vote lets us infer. Say that 52% are located at party
A, and 48% at party B, and that WLOG, A's location on the line is closer
to 0 than is B's. Then the count starts at A, and proceeds as such,
electing candidates from A's list until A has its share, at which point
it jumps closer to B. In essence, party list becomes a equal-rank
plurality version where everybody either votes A1 = ... = An or B1 = ...
= Bn.
The problem of synthesizing the distribution on the political line still
remains, though. If people vote rationally (that is, that there is no
noise), one can get some extent of the way by observing
eee A fff ggg B hhh iii C jjj
0|-----------------------------|1
The e faction would vote A > B > C. So would the f faction, while the g
faction would vote B > A > C and the h faction, B > C > A. The i and j
faction votes C > B > A. Noise votes are A > C > B and C > A > B.
However, while this gives us some information as to the relative sizes
of the factions, it does not tell us whether (say) the e faction is
large because there's a peak of support there, or because the others are
more extreme, like this:
eeeeeeee A ff gg B hh ii C jj
0|-----------------------------|1
In the case of Condorcet, it doesn't matter, since Black's
single-peakedness theorem says that in the one-dimensional case with
voters preferring candidates closer to them, there'll always be a CW and
that CW is the candidate closest to the median voter.
Can we use this to make "multiwinner Condorcet" where the k-ile property
holds? To do so, I would have to understand the aforementioned
single-peakedness theorem to know how it works, which I don't.
If I were to make a guess, I think it's because Condorcet removes the
other candidates from each pairwise check. Assume A is the median
candidate. Then on A vs B, A is closer to more voters than B is. If that
is all that's needed, then we could imagine a Condorcet analog where if
a voter is closer to A or B than to C or D, it counts as a win for {A,
B}. If a council {X,Y} is a CW in this way, and X is closer to 0 than is
Y, is then X closest to the 33% position, and is Y closest to the 67%
position? I don't know.
In any case, even if I'm right about the above, we have to figure out
what "{A, B} is closer than {C, D}" means. If a voter ranks A > B > C >
D or B > A > C > D, it's pretty obvious that {A, B} is closer. But what
of A > C > B > D or C > A > D > B ? And is it possible to make a system
that picks the (k+1)-ile closest candidates without having to go through
all possible combinations of the council in the worst case?
-
This may have been a bit meandering, but I wrote it as I went on. My
general idea is this: multiwinner election is not as well defined as
single-winner election, so I tried to find a way of defining it better
by referring to issue space, and in a way so that it reduces to
Condorcet in the single-winner case. Then I wrote a bit about the
limitations of this (that we can't infer the shape of the curve in issue
space from ranking alone), and that perhaps we don't need the shape of
that curve - but on that, I'm uncertain, since I don't know Black's
theorem.
There's also the problem of noise and multiple dimensions to consider,
but let's keep this simple :-)
Any ideas, replies?
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