[EM] Multiwinner unanimity criteria

[Kristofer Munsterhjelm]

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.