Aaron Armitage wrote:
Perhaps the voter is given an extra vote to augment his more strongly
held preferences, so that if he gives it all to his first preference,
that candidate gets two votes against all other candidates, but the
second choice gets one vote against everyone ranked lower. On the other
hand, if he gives half to his first choice and half to his second, then
the second choice gets 1.5 against third and lower candidates, but the
first gets 1.5 against the second and 2 against third and lower. If he
gives it all to third, then the top three get 2 against everyone lower,
but the preferences first > second > third all get 1, as does fourth >
fifth. And so on. This would be more complicated and involve some
interesting strategic choices. At first glance it would seem optimum to
treat it as an approval cutoff. At least it would avoid the arbitrariness
of assuming that the first vs. second preference is more important than
second vs. third, and that by the same multiplier for every voter.
The endpoint of that line of thought, I think, is Cardinal Weighted
Pairwise. The input is a rated (Range-style) ballot. Say, WLOG, that A
is more highly rated than B. Then A beats B by (rating of A - rating of
B). So, for instance, if on a 0-100 ballot:
A (100) > B (75) > C (20)
you get
A > B by 25
A > C by 80
B > C by 55.
Use your favorite method to find the winner, as CWP produces a Condorcet
matrix that can be used by any method that employs the matrix alone
(e.g, not Nanson, Baldwin, or similar).
If you want to vote nearly Approval-style, you would do something like
A (100) > B (99) > C (98) > D (2) > E (1) > F(0)
but that may not be optimal strategy; one could argue that in the same
way that ranking Approval style is not optimal in ranked Condorcet
methods, rating nearly Approval style isn't for CWP.
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