I had a post lined up about my second approach to try to find a monotone
resistant set method but failing, and then I thought I would do a few
more tests to just add supplementary information...
... and then that turned into a much longer and more involved
experiment, and I think I've done it. I think I've found a monotone
resistant set method.
I want to prove it before I claim it with certainty, but my simulations
have been unable to find any counterexamples yet.
So either I have been thoroughly trolled, or I beat the troll. I'll find
out one way or the other :-)
But another observation: In 2023, Filip Ejlak commented that
Resistant//X methods fail monotonicity[1]. I responded by saying that
this means that a hypothetical "independence from non-resistant
alternatives" property (independence from vulnerable alternatives,
maybe?) is incompatible with monotonicity.
But I think we can say something more general: if a method elects from
an "X-set" (of some definition) and passes majority, then independence
from X-dominated alternatives is impossible if there exist situations
where set X contains only a single candidate A, but someone outside the
set (say candidate C) beats A pairwise. In that case we can eliminate
everybody but A and C. By independence from X-dominated alternatives,
this shouldn't change the outcome, but C beats A pairwise and the method
passes majority, it is forced to elect C. Hence independence from
X-dominated alternatives is impossible.
And that's what's really going on in Filip's example. A is the sole
resistant set member because A~>B (A has more than a third of first
preferences and also beats B pairwise), and B~>C (ditto B and B beats C
pairwise). However, C beats A pairwise. So no majoritarian method can
pass independence of non-resistant alternatives because if it did, it
must elect A in Filip's example, but then we eliminate B (which is not
in the resistant set) and it must elect C.
-km
(65e733d0c246a08410c8b9330abb...)
[1]
http://lists.electorama.com/pipermail/election-methods-electorama.com/2023-August/004753.html
I had a post lined up about my second approach to try to find a monotone
resistant set method but failing, and then I thought I would do a few
more tests to just add supplementary information...
... and then that turned into a much longer and more involved
experiment, and I think I've done it. I think I've found a monotone
resistant set method.
I want to prove it before I claim it with certainty, but my simulations
have been unable to find any counterexamples yet.
So either I have been thoroughly trolled, or I beat the troll. I'll find
out one way or the other :-)
But another observation: In 2023, Filip Ejlak commented that
Resistant//X methods fail monotonicity[1]. I responded by saying that
this means that a hypothetical "independence from non-resistant
alternatives" property (independence from vulnerable alternatives,
maybe?) is incompatible with monotonicity.
But I think we can say something more general: if a method elects from
an "X-set" (of some definition) and passes majority, then independence
from X-dominated alternatives is impossible if there exist situations
where set X contains only a single candidate A, but someone outside the
set (say candidate C) beats A pairwise. In that case we can eliminate
everybody but A and C. By independence from X-dominated alternatives,
this shouldn't change the outcome, but C beats A pairwise and the method
passes majority, it is forced to elect C. Hence independence from
X-dominated alternatives is impossible.
And that's what's really going on in Filip's example. A is the sole
resistant set member because A~>B (A has more than a third of first
preferences and also beats B pairwise), and B~>C (ditto B and B beats C
pairwise). However, C beats A pairwise. So no majoritarian method can
pass independence of non-resistant alternatives because if it did, it
must elect A in Filip's example, but then we eliminate B (which is not
in the resistant set) and it must elect C.
-km
(65e733d0c246a08410c8b9330abb...)
[1]
http://lists.electorama.com/pipermail/election-methods-electorama.com/2023-August/004753.html
