[EM] Fixing IRV

[Blake Cretney]

On Wed, 08 Aug 2001 17:36:27 -0700
Richard Moore <rmoore4 at home.com> wrote:

> Markus Schulze wrote:
> > Dear Richard,
> > 
> > you wrote (7 Aug 2001):
> > 
> >>Actually, IIRC, there is a proof somewhere in the literature
> >>that elimination methods are not monotonic. Does anyone recall
> >>the theorem I mentioned above about elimination and monotonicity?
> >>
> > 
> > Some elimination methods are monotonic (e.g. Ranked Pairs).
> 
> Since when is RP considered an elimination method?

RP gives you a complete ordering of the candidates.  This ordering
gives you a lowest candidate.  So, you might suggest a method,
RP-elimination, that finds the candidate when you successively
eliminate the lowest RP-ranked candidate.  Who is that candidate? 
Turns out, it's the same as the normal RP winner.  So, RP-elimination
= RP.  Since RP is monotonic, RP-elimination must be as well.

> If we have
> 
> 6 
> ABC
> 5 
> CAB
> 4 
> BCA
> 
> then RP gives the following rankings for the pairwise contests:
> 
> 11 
> A>B
> 10 
> B>C
> 9 
> C>A
> 
> with A as the winner. If RP were done as an elimination 
> method, then B would be eliminated following the highest 
> pairwise defeat. We then wouldn't bother comparing B and C, 
> and we would get

The complete RP ranking is A>B>C.  So, you first eliminate C.  This
gives A vs. B.  A is the RP winner between them. 

Now, you might rightly state that although I could define RP as an
elimination method, it would be ridiculous to do so.  Nevertheless,
since I could, it follows that there can't be a proof that no
elimination method is monotonic, since this isn't technically true.

---
Blake Cretney