[EM] Simple monotonicity question

[Kristofer Munsterhjelm]

Juho wrote:
>> does a candidate-elimination method have to be able to eliminate the 
>> Condorcet winner in a three-candidate scenario in order to be 
>> nonmonotonic with only three candidates?
> 
> 
> I'm not quite sure what the intended question is but isn't it enough to 
> eliminate the candidate with most first preference votes unless it is 
> the Condorcet winner. This method will be non-monotonic when there is no 
> Condorcet winner.

I'll try to rephrase it:

Limit yourself to the case of only three candidates. If the method works 
by eliminating candidates until there is only one left, and it never 
eliminates the CW when there is one, must it then be monotone?

I guess it might still exhibit nonmonotonicity if it's possible to have 
a situation with no CW, then raise some candidate and still have no CW. 
Consider a contrived method that eliminates candidates with the most 
first preference votes. Then shouldn't raising some candidate to first 
place make it lose, and shouldn't that be nonmonotone if there is no CW? 
Hm. If it's possible to construct such a situation, that is...

If "it never eliminates the CW when there is one" is limited to the 
original CW, then that might be possible. On the other hand, if it 
"never eliminates the CW when there is one" among the not-yet-eliminated 
subset, then when there is only two candidates left, it must pick the 
one with the most first preference votes (because that one is the CW 
among those who remain).

An easy one-way check would be to see if it's possible to make a 
nonmontonicity scenario for BTR-IRV with only three candidates. If it 
is, then my question has been answered with that three-candidate 
elimination methods that don't eliminate the CW can still be 
nonmonotonic. If it isn't, however, the question's still open, because 
there might be some contrived method where one can force nonmonotonicity.