[EM] No. Condorcet and Hare do not share the same problem with computational complexity and process transparency.

[Kristofer Munsterhjelm]

On 2024-03-27 03:15, Closed Limelike Curves wrote:
>>  "Strong monotonicity" is my term for the stronger mono-raise where
>>  raising A can't change whether the outcome ranks some other B ahead of
>>  C. Range passes it, for instance, but it's very rare in ranked methods.
> 
> Am I misunderstanding this, or is it just independence of irrelevant 
> alternatives?

It's very close, but I think IIA is slightly stronger. That is, for 
every reasonable method they coincide, but you can contrive examples 
where they don't.

First, strong monotonicity requires ordinary monotonicity. The perverse 
Range version where the lowest score candidate wins passes IIA but not 
monotonicity.

Second, you could have a method where just adding a candidate ranked 
below everybody else perturbs the ranking of the other candidates, 
perhaps something weird that has a quota where the number of candidates 
is a term. You'd expect every reasonable voting method to pass 
"independence from universally least-ranked candidates", but strictly 
speaking if the method doesn't pass it, you could have strong 
monotonicity without IIA. And there could be a very slim chance that 
such a method would be useful as an elimination order component of a 
larger method, at least for a proof of concept... though I doubt it.



Tangentially, here's a fun trick related to strong monotonicity or IIA.

Suppose you have a single-winner elimination method that takes a base 
method order, and has some logic like:
	- If [some special case based on the remaining candidates] then X wins.
	- Otherwise, eliminate the lowest ranked continuing candidate and go 
back to the previous step.

And suppose that this method is monotone if the base order passes IIA. 
Then you can make a monotone method like this:
	- For each ballot b:
		Let W(x) be the winner of the elimination method when b
			is used as the base method order
		Increment the count of W(x) by one (or the ballot's
			weight)
	- Elect the candidate with maximum count, i.e. the Plurality
		winner according to the social orders produced by the
		elimination method.

This works because dictatorial methods pass IIA, and Plurality itself is 
monotone. I was hoping that I could use the resistant set elimination 
method I described a while ago (in the "not UD" post) to create a 
monotone Resistant method this way. But it turned out that the 
elimination method wasn't actually monotone, not even with the input 
order fixed.

If you do this with loser elimination with a majority check, you get a 
method that passes Condorcet loser (because the C.L. is always 
eliminated). But I'm not sure what its other properties would be. 
Similarly with BTR-type or Benham-type elimination you get Condorcet, 
but I'm not sure if it preserves monotonicity as an inner method.

It might be possible to apply this trick to the preset elimination STV 
method of my previous post, but again I'm unsure if the "monotone if the 
input order stays the same but someone's raised in the election itself" 
property holds. In that case, the "winners" to do Plurality over would 
be sets of candidates. However, it's possible that vote-splitting would 
destroy the outer method's monotonicity, e.g. {A, B, C} has the max 
count of 10, then when you raise A, you get {A, B, C} with a count of 9 
and {A, B, D} with a count of 8.

In any case, Schulze STV seems to pass monotonicity - it hasn't been 
proven, but nobody has been able to find a counterexample either. So in 
that sense, we already have a monotone proportional ranked multiwinner 
method.

-km