[EM] "No splitting rule"

[David Catchpole]

Hey guys- here's two types of single-winner no-splitting rule. They
attempt to disallow additional candidates splitting the vote and
disadvantaging their sides- however, they do allow additional candidates
helping the chances of other candidates "on their side"

Rule 1
Of those who prefer an additional candidate over the old winner, at
least as many prefer the new winner over the old winner as prefer the old 
winner over the new winner, or there's no change.

(With respect to a removal of a candidate)
With the removal of a candidate A, either-

(i) A was the old winner; or
(ii) There's no change; or
(iii) Of those who rank A>NW, at least as many rank OW>NW as NW>OW: at
least as many (A>OW>NW)+(OW>A>NW)+(OW=A>NW) as (A>NW>OW)

Now can anybody see what's offensive about Rule 1? No, it's not (iii),
that's just a statement of admission (People might do worse not to run Mr.
A- but you see, they never do worse running him, do they?). It's something
to do with who comprises the "yeah, we like OW" A>NW group. It includes
people who rank OW>A, the OW hacks who might overwhelm the numbers of the
A>NW>OW even though really a majority of A voters preferred NW.

Rule 2 is slightly more restrictive in that it states that-

of those who rank the additional candidate over both the old and new
winners, at least as many prefer the new winner over the old winner as
prefer the old winner over the new winner, or there's no change.

(With respect to a removal of a candidate)
With the removal of a candidate A, either-

(i) A was the old winner; or
(ii) There's no change; or
(iii) Of those who rank A>{OW,NW}, at least as many OW>NW as NW>OW: At
least as many A>OW>NW as A>NW>OW.

An extension of rule 2 into multiple winner elections (with respect to
removal)

With the removal of a candidate A, either-

(i) A was one of the old winners; or
(ii) There's no change; or
(iii) Of those who rank some possible combination of winning candidates
including A over the old and new winning combinations, at least as many
prefer the old winning combination to the new winning combination as
prefer the new winning combination to the old winning combination (whew!)