# [EM] Grouping/bifurcation elimination methods

Richard Moore rmoore4 at home.com
Mon Aug 13 18:25:02 PDT 2001

```Roy wrote:
> Does anyone know whether the theorem that all elimination methods
> suffer failure of monotonicity applies to one-at-a-time elimination
> only, or would include any kind of elimination?

Since I haven't found a formal statement of this theorem on
the Internet I guess I'll have to make a trip to the library
one of these days, to try to locate Riker's book.

Here's something that occurred to me that relates to the
above question.

If we use pairwise single elimination, with the candidates
in a predetermined starting order, then the following seem
to be true:

1. The method is Condorcet compliant, since a CW cannot be
beaten by any other candidate;
2. The method is Smith compliant, since a Smith set member
can only be beaten by another Smith set member;
3. The method is monotonic, since improving the winner's
support over another candidate cannot change the winner to a
loser.

Though I could be mistaken, I don't see how this method
could be implemented as a non-elimination method.

Now, if instead of using a predetermined starting order, we
seed the candidates based on information derived from the
ballots (such as Plurality or Borda rankings), then the
method can be nonmonotonic. For now, it becomes possible in
certain cases to change the seeding by swapping to
adjacently ranked candidates on one ballot. Once the seeding
order is changed, a Smith set candidate who originally was
eliminated early might now survive and defeat the original
winner.

So it does seem likely that Riker (or Cranor) meant
something narrower than what I (and perhaps others on this
list) think of as "elimination". Perhaps the theorem does
apply only (as Roy suggests) when a single candidate is
eliminated in each round. Or perhaps it doesn't apply in
single elimination with predetermined starting order because
information (or is it noise?) outside the balloting is used
to seed the candidates.

Richard

```