[EM] Method X

[Kevin Venzke]

Hi Kristofer,

It wasn't so easy, but regrettably I think I have a monotonicity counter-example:

408: B>C>A
329: A>C>B
126: C>A>B
91: C>B>A
43: A>B>C  -->  B>A>C
(total 997)

For the first round, A and B votes both exceed 1/3rd (332.33) and so only C can be
eliminated.
The match-up A:B gives B a very slight win of 499 vs 498 for A. C can't score anything.
Scores: B 499, A 498, C 0.

Now change the 43 to B>A>C, theoretically helping B further.
First round totals become 329 A, 451 B, 217 C. So it is now allowed to eliminate A.
Both A and B fare worse against C than against each other and so prefer to score off of
eliminating C.
B improves its score to 542 while A's score is reduced to 455.
However, when A is eliminated, C can score 546 from their matchup with B.
New scores: C 546, B 542, A 455.

Let me know if you think I have the method wrong.

One thing I noticed is that modifying the quota rule allows you at one extreme to
implement IRV (i.e. by saying that only the candidate with the fewest votes can be
eliminated each round) and at the other extreme to implement "max votes-for wins" (by
imposing no quota requirement at all). While the latter is monotone, it doesn't satisfy
majority favorite. It's interesting to consider whether any quota rule could at least
preserve monotonicity and add majority favorite. I'm thinking no, though.

Kevin
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