[EM] Does it satisfy Droop proportionality, monotonicity, and clone independence?
Ross Hyman
rahyman at sbcglobal.net
Thu Jan 2 06:36:06 PST 2014
I think the method could be monotonic because although QPQ and STV are not monotonic in general, we only use them for the case of n+1 candidate elections for n seats. Maybe these methods or another have the desired monotonicity property in that restricted case.
The property we desire is that for an n+1 candidate election for n seats, raising one of the n winners on some ballots cannot cause that winner or one of the other winners to lose.
So if one candidate, say X, is raised on some of the ballots, and a monotonic social ranking is used so that the only effect on the social ranking is to raise X, then if X was a winner previously it should remain a winner. This is so because for it to have been a winner previously, it had to have been a winner of every n+1 election that includes it and winners of lower rank. Provided the same winners of lower rank still win when X is raised (which is the case if the desired monotonicity property is obeyed for n+1 candidate elections for n seats) , X will win again.
________________________________
From: Kristofer Munsterhjelm
On 01/02/2014 05:25 AM, Ross Hyman wrote:
> If the societal ranking that makes it up is monotonic and clone
> independent, which is the case for Schulze beat path and Tideman ranked
> pairs, then does the combined method satisfy multi-winner
> generalizations of those single winner properties?
> If it can be shown that the method satisfies Droop proportionality,
> monotonicity, and clone independence, then I think it would be
> preferable to conventional QPQ or STV. Otherwise not.
I could try digging up my election simulator code and see if I could test the multiwinner methods for clone independence. As far as I recall (from Woodall), we don't know of any Droop-propotortional cloneproof methods. While IRV is cloneproof, STV is not.
By monotonicity, in multiwinner methods I think there are two kinds of monotonicity, which I called strong and weak monotonicity.
Weak monotonicity goes like this: If X is in the outcome, and someone ranks X higher, then X should not drop out. Conversely, if X is not in the outcome, and someone ranks X lower, then X should not win.
Strong monotonicity is like this: if a voter raises any subset of the winners in his rankings (not necessarily by the same amount), leaving the rest alone, then none of the raised candidates should drop out.
For an example where strong monotonicity constrains but weak does not, consider a situation where A and B are among the winners. Then someone who ranked C>A>B changes his ballot to vote A>B>C and B loses his seat. That would not be a weak monotonicity violation (since the ballot didn't just raise A or B), but it would be a strong one.
Strong monotonicity is *very* strong, because it implies that if a winning candidate is raised, nothing should change at all in the outcome (since it can be considered an instance where all the winners are raised, but all but one is raised no steps).
Which kind of monotonicity did you have in mind?
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