[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 

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 

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.


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