Here are two multiwinner election criteria:
1. Weak unanimous subelection criterion.
If the election is for p seats, and a set of the same (p-1) candidates
are ranked above all others by every voter, then the method should pick
those (p-1) candidates and elect the remaining candidate by the
single-winner election case of its own method, where the (p-1) already
elected candidates are ineligible to win.
2. Strong unanimous subelection criterion.
If the election is for p seats, and a set of the same k candidates, k <
p, are ranked above all others by every voter, then the method should
pick those k candidates and elect the remaining candidates by the
(p-k)-seat multiwinner election case of its own method, where the k
already elected candidates are ineligible to win.
Do these seem reasonable? Every elect-and-punish method would presumably
pass, since after electing the unanimous favorite, they would punish
every voter equally and so the ballots' relative strength don't change.
But consider a method that uses Sainte-Lague type reweighting, where the
"punishing" depends on how many candidates this voter has already
elected. Then if we have a 100-winner election and 95 are unanimously
elected, then the voters who elect the next candidate will only be
reweighted down by 1/192, which is much less than the 1/2 that would be
the case for the reduced 5-winner election with the 95 being ineligible.
So maybe it's not a good thing after all? I'm not sure. With multiwinner
elections, effects are subtle and they can be quite unintuitive.
We would also have to define "candidates in question are ineligible to
win". That either means that the candidates are eliminated outright, or
that the election proceeds as the single-winner instance, and the
candidates are removed from the output social ordering. Which one would
make most sense? It may even be that when the candidates set ineligible
are all unanimously preferred to the rest, the two ways of removing them
give the same result.
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