[EM] Does it satisfy Droop proportionality, monotonicity, and clone independence?

[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?