I was thinking recently about how one might design a method aimed to
minimize potential for regret at least for supporters of the median
candidate. By "regret" I mean especially the situation that supporters
of the median candidate give the election away by voting also for a
I had an idea worth sharing, I think...
On cast approval ballots I like to guess that the median candidate is
the one to whom the greatest opposition is the least. (The greatest
opposition to a candidate X is defined as the size of the largest group
of voters who approve a common candidate and disapprove X. In other
words, if you remove all ballots approving X, what is then the highest
approval score of any candidate?)
I like this measure because typically supporters of the median candidate
(when there is also a "left" and a "right" candidate) can't hurt this
candidate by also approving the "left" or "right" candidate that is their
second choice. It only hurts the median candidate sometimes when the
greatest opposition to the second choice is the median candidate (so
that this second choice is turned into the median candidate). But
usually we'd expect that the greatest opposition to the second choice
is coming from the opposite side of the spectrum, not the median.
The trouble with always electing this "median" candidate is that he might
have very little approval:
B would be the "median" candidate with just 2 approval.
Assume that approved candidates would have been ranked if a rank ballot
had been used. Assume also that disapproved candidates would not have
been given any ranking. Given these assumptions, a candidate in an
approval election might have been the Condorcet winner if and only if
his approval score is higher than the greatest opposition to him.
So this compromise occurred to me:
"Elect the candidate to whom the greatest opposition is the least
(breaking ties in favor of greatest approval), whose approval is at least
as high as the greatest opposition to him."
I don't have proofs, but simulations of mine couldn't find any monotonicity
or FBC failures with this. (Actually I first tried "elect the candidate
with the least max opposition IF his approval is at least as high as etc.,
otherwise elect the approval winner," but this had both problems.)
Compared to Approval this makes a difference in a scenario like this:
Approval scores are A 55, B 40, C 45.
Max opposition scores are A 45, B 30, C 55.
Approval elects A. This method ("ALMO") identifies B as closest to
"median" and sees that B has enough approval to possibly be the Condorcet
winner on rank ballots, and so elects B.
(Naturally you can argue that this isn't an improvement, or that
"opposition" isn't a useful concept.)
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