[EM] Fixing IRV
Blake Cretney
bcretney at postmark.net
Wed Aug 8 19:44:48 PDT 2001
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
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