[EM] Sequential elimination method that turns out to be a generalization of fpA-fpC

[Filip Ejlak]

Ok, after trying out a few things I found something that looks promising in
terms of passing all of ISDA, DMTCBR, Cloneproofness and, I hope,
Monotonicity. The scoring rule is similar to Friendly Cover (but without
subtracting fpC); the elimination rule is a bit atypical.

The definition:

"*Let X's score be the sum of 1st preferences of candidates that are
pairwise beaten by all candidates that pairwise beat X. In each round
eliminate such a candidate that the sum of the eliminated candidate's score
and the greatest score growth caused by the elimination is minimized*."

The growth-minimizing part works for at least two reasons: 1) It reinforces
the behaviour we would expect in a situation with no cycles, when an
elimination shouldn't cause a positive score growth (if an eliminated
candidate was worse than X, then its 1st preferences were already included
in X's score). 2) In cycle situations it should benefit candidates that
already have a high score (because letting a low-score candidate achieve
some score level is less preferable to letting a high-score candidate
achieve such level).

Also notice that the eliminated candidate's score is equal to the greatest
score decrease caused by the elimination, so there is a nice symmetry
between these two things we sum. It also means we can express the
elimination rule in a more elegant way ("*eliminate such a candidate that
the greatest score distance change between any two candidates is minimized*"),
but perhaps the former definition makes it more clear why it works.

It seems to me that the incentive to eliminate a non-Smith candidate
because of a low score should always outweigh the incentive to eliminate a
Smith candidate because of a low score growth, so all non-Smith candidates
should be eliminated first.

Proof that the 3-candidate case is equivalent to fpA-fpC:

If there is no cycle, it is trivial as both methods are Condorcet. If there
is an A>B>C>A cycle, then we look at the possible eliminations. Let "a" be
the number of A's 1st preferences, "b" of B, "c" of C.

- if C is eliminated, A's score increases by b+c and A wins with B

- if A is eliminated, B's score increases by a+c and B wins with C

- if B is eliminated, C's score increases by a+b and C wins with A

We add eliminated scores to score growths in order to find the optimal
elimination. If 2c+b is the lowest, A wins. If 2a+c is the lowest, B wins.
If 2b+a is the lowest, C wins.

Notice that min(2c+b, 2a+c, 2b+a) = min(2c+b-(a+b+c), 2a+c-(a+b+c),
2b+a-(a+b+c)) = min(c-a, a-b, b-c) = max(a-c, b-a, c-b). So we get the
fpA-fpC formula, QED.
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