I'm coming around to the conclusion that reverse symmetry is not such a
good idea in multiwinner elections if one's aim is proportional
representation.
Consider the following examples:
Example 1:
25 ABC
25 BAC
25 CAB
25 CBA
In this example if we reverse the preferences, we get the same number of
each preference as before, so the candidate subsets stay in the same
order. If the method is such that it reverses the order, then the only
possibility is that all subsets are tied. In particular, {A,C} is tied
with {A,B}. This means that even with 50% of the first place vote, C
isn't guaranteed a seat in a two winner election.
Example 2.
50 A>B=C=D=E=F
50 B=C=D=E=F>A
Again, reversal of all preferences yields exactly the same number of each
type of ballot, so the winning order of the subsets must be unchanged, and
also reversed if the method is symmetrical. So again, all of the subsets
are tied. In particular, {B,C,D,E} is tied with {A,C,D,E} even though A
is the only approved choice of half of the electorate.
The only way to fair proportional representation seems to be through
abandoning the symmetrical method that I have been playing with recently.
The lower median m is out, along with the j1 and j2.
The upper median M, and the k1 and k2 are adequate by themselves.
Doubling the k values gets rid of the daunting integral.
The margin is just
1+1/2+...+1/(2*k1) - 1+1/2+...+1/(2*k2) .
to be continued ...
Forest
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