[EM] Helping a candidate in the case of ties

[Kristofer Munsterhjelm]

Dave Ketchum wrote:
> Of course, you have to read the voter's mind to know if the change might 
> have been seen as desirable.
> 
> I was into tactics.

So was I. Consider the monotonicity (mono-raise) criterion, which is 
defined as: "raising a candidate x on some ballots on which x is listed 
should not harm x". Raise is defined as moving the candidate to a higher 
rank, and "harm" is usually defined in the context of winners, for 
monotonicity, that some winning candidate A is harmed by a change if he 
used to be the winner but no longer is.

I thought it would be interesting to see if some methods that pass 
ordinary monotonicity with respect to the winners fails it with respect 
to the ordering. Of course, that means you'd have to have a definition 
of harming X that works for all places in the order (except last).

For a strict ordering (nobody shares first, second, third, etc), it's 
easy. X is harmed by a change in the ballots if the outcome used to rank 
X at pth place, but now ranks X at qth place, and q > p. For instance, 
if A is raised and thus A goes from second place to third, the method 
fails the extended monotonicity criterion.

The problem is that I don't see how to define "harms X" in a way that 
both seems sensible and resolves to the "goes from pth to qth place" in 
the case of strict rankings.

Actually checking for whether a method fails monotonicity remains the 
same: take a list of random ballots, raise a certain candidate on some 
of these, and see if that candidate is harmed. It's also possible to do 
it the other way: lower the candidate on some ballots and see if that 
candidate is helped. Whatever the voters' minds, the check relies only 
on the two elections (with modified and unmodified ballots) and the 
outcomes of those elections according to the method being tested.

> One thought I had was a base from which to think of more-or-less 
> controllable changes:  Start with equal size parties and all members 
> doing bullet voting - result is a tie with all candidates getting equal 
> votes for and against.

How do you mean?