I just thought of a possible Approval variant of M-Set Webster (my
"nameless" monotone multiwinner method). The only change that is
required is the construction of the sets that are used as a basis for
the constraints. This change is as follows:
Each voter gives one point to every subset of the candidates they approved.
The rest of the method is run as usual: the constraint phase determines
the least divisor for which at least one council passes those
constraints, and the margins phase determines which of the councils that
pass that will actually be elected.
It seems to be both proportional in the multiwinner case and reduce to
Approval in the single-winner case, as the only constraint that matters
in that case is "one or more of these", and the first single-candidate
set to limit that is the set of the Approval winner.
The only situation where that can't elect the Approval winner would be
if some other set (which must be of multiple candidates) were to impose
a constraint away from the Approval winner. However, I can't see any way
to construct such a multiple candidate set, because all voters who
contribute to the set also contribute to each candidate within the set,
thus increasing the chance that one of the set members are going to be
the Approval winner.
Multiwinner proportionality is easier to show: if one group consistently
votes for a certain subset, and another for another, then the constraint
will impose that at least a great fraction of the subset as the fraction
of the group with respect to the voters, will be from that group -
unless the constraints then become impossible to meet.
However, because the method gives one point to every possible subset,
the number of subsets can become very large indeed - by approving of
every candidate, a voter may force the method to generate 2^n subsets
(where n is the number of candidates). This is in constrast to proper
solid coalitions where there can be no more than v*n^2 coalitions (where
v is the number of voters).
Perhaps it's possible to use additional structure to do something about
it, but unless that is done, the method won't be very practical in the
case of many candidates. This is unfortunate, because I was considering
the use of multiwinner Approval methods with large council sizes as a
"nomination phase" for ranked voting where the number of potential
candidates may be very large. The idea would be to let people choose,
approval style, from a large group (or nominate their own choices), then
to use these ballots to find a smaller group about which one could gain
more detailed preference data from the voters without overloading the
voters themselves.
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