[EM] Considering teaming criteria for multiwinner methods

[Kristofer Munsterhjelm]

In Voting matters issue 3
(http://www.votingmatters.org.uk/ISSUE3/P5.HTM), Douglas Woodall
commented that clone-no-harm (invulnerability to teaming) is not a
desirable property, and is incompatible with the DPC.

It would be pretty easy to modify the teaming property to respect the
DPC, but then only multiwinner systems that pass the DPC would pass that
criterion. This would be a problem for say, a multiwinner method that
reduces to a divisor method (which can fail quota).

So the other day I thought of a very weak clone-no-harm criterion that
still seems to discard too majoritarian methods while not taking a
position on any kind of quota one might use. It is:

Weak teaming resistance / weak clone-no-harm: Let X be any candidate
that is not unanimously voted first. Then for every such X, there must
exist a number of seats s so that clones of X do not occupy all s seats
no matter how many clones X is replaced with.

Trying to use Ranked Pairs as a proportional method by taking the first
s winners (first place, second place, etc) as the council immediately
fails this criterion. More generally, trying to directly extend any
single-winner cloneproof method that passes LIIA will also fail. Proof
by contradiction: Suppose that the election for s seats is one that
passes the criterion above. Let X be the winner of the single-winner
case. Now clone X s times. Because the base method is cloneproof, this
cannot make X lose, so one of the X-clones must be ranked first in the
outcome. And because the base method passes LIIA, all the other clones
must be ranked right after X. Thus all s highest ranks are occupied by
clones of X. But that contradicts the assumption that s proved weak
clone-no-harm to be satisfied.

(Similar proofs can be had for anything passing Smith and independence
of clones.)

Anything that passes some kind of quota will pass this WCNH, because at
some number of seats, the non-clones will have more than a quota's worth
and thus be entitled to a seat. At that point, it doesn't matter how
many clones of X there are, because the quota criterion will demand that
not all the seats go to clones of X.

The clone criterion might be too weak. Even something like SNTV would
seem to pass it. If we want something closer to single-winner clone
independence (or clone-no-harm), we would want something that limits the
effects of cloning within each quota situation. A single-winner method
can pass mutual majority and still fail clone independence, as long as
the candidate benefitting is part of the smallest mutual majority set.
The hard part is to make teaming resistance work while not referring to
quotas, the way single-winner teaming resistance works without referring
to mutual majority.


Another weak clone criterion would be: for any s, replace every
candidate with s clones each. Adding even more clones of X should
neither help nor harm X, and should not help any other Y.

That's a hack to avoid the situation where teaming should be beneficial
(when all the clones are supported by a sufficiently large fraction of
the voters that they should be elected).

In a party list setting where each party is considered to have enough
candidates to fill the council, that's what you get, and party list
methods should not have a problem; we could just apply standard clone
independence checks to them.

A more narrow set of party list methods would take ranked ballots as
input and produce an n-tuple of (party name, score); and then these
scores would be fed into something like Webster's method to produce an
allocation. For that kind of method, all three of Woodall's clone
criteria make sense: the score of some candidate X relative to the other
candidates should not increase nor decrease if X is cloned, or if some
other Y is cloned.