[EM] General PR question

[Andy Jennings]

Forest and I were discussing PR last week and the following  situation came
up.  Suppose there are five candidates, A, B, C, D, E.  A and B evenly
divide the electorate and, in a completely orthogonal way, C, D, and E
evenly divide the electorate.  That is:

One-sixth of the electorate approves A and C.
One-sixth of the electorate approves A and D.
One-sixth of the electorate approves A and E.
One-sixth of the electorate approves B and C.
One-sixth of the electorate approves B and D.
One-sixth of the electorate approves B and E.

It is obvious that the best two-winner representative body is A and B.  What
is the best three-winner representative body?

CDE seems to be the fairest.  As Forest said, it is "envy-free".

Some methods would choose ABC, ABD, or ABE, which seem to give more total
satisfaction.

Is one unequivocally better than the other?

I tend to feel that each representative should represent one-third of the
voters, so CDE is a much better outcome.  Certain methods, like STV, Monroe,
and AT-TV (I think) can even output a list of which voters are represented
by each candidate, which I really like.

I also note that if there was another candidate, F, approved by everybody,
it is probably true that ABF would be an even better committee than CDE.  Is
there a method that can choose CDE in the first case and ABF in the second
case?

Andy
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