A way to reconcile majority rule and PR with even numbers of winners: Break ties in PR legislatures in favor of the side collectively preferred by a majority of voters/that gives greater overall utility

If you have a vote split in a 6-winner election between two parties of:
51 A
49 B
the proportionally better answer seems to be 3 A 3 B but this contradicts majority rule, which would seem to require 4 A 2 B, since otherwise the A’s and B’s will tie when voting contentiously in the legislature and the majority of voters won’t have full control (though they’ll have veto power). A similar analogy could probably be made with cardinal ballots if one party gave greater social utility overall.
If the tiebreaker suggested above is used (and for cardinal methods, it might make sense to say that one side must not only give greater overall utility, but receive a majority of support i.e. majority of points possible, which I suspect can be thought of as receiving majority preference after applying the KP transform), then the majority will have full control even under the proportionally better result.

Maybe when computing majority preference, votes for irrelevant candidates shouldn’t be figured in. On the other hand, those votes may be protest voters who wanted to make it harder for anyone to get a majority.

But you’ll always get problems like this unless the number of seats can be divided exactly proportionally to the votes, which in practice will never happen. Similarly yould could have:

26: A1
25: A2
49: B

A1 and A2 aren’t officially affiliated with each other in any way, but I’ve called them that because they could have very similar policies and this is effectively a vote split. So should A1 and A2 between them still get 4 out of the 6?

In summary, I don’t think the majority/non-majority knife-edge is any more relevant in practice than any case where a party gets just above or below a particular quota.

I suspect in my jumble of words you may have missed

i.e. if those 51 voters marked down that they prefer the set of {A1, A2} over B (or if those candidates had declared an official pre-election coalition), then even if A1 and A2 only get 3 out of 6 seats, any ties where A1 and A2 are on the same side in the legislature can be broken in their favor because they are majority-preferred.

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