[EM] MinLV(erw) Sorted Margins Elimination
C.Benham
cbenham at adam.com.au
Thu May 4 01:24:53 PDT 2023
I made a strange mistake in the working out of my first example (but not
the result). Below it is corrected.
(I actually first made the mistake in 2016 and I soon noticed it then
and posted a corrected version, but what I posted last month
was mostly copied from the initial uncorrected post. )
Chris Benham
My favourite method that meets both Condorcet and Chicken Dilemma is
'Min Losing-Votes (equal-ranking whole) Sorted Margins Elimination':
*Voters rank from the top whatever number of candidates they like.
Equal-ranking and truncation are allowed.
For the purpose of determining candidates' pairwise scores:
a ballot that votes both X and Y above no other (remaining) candidates
contributes nothing to X's pairwise score versus Y and vice versa,
a ballot that ranks X and Y equal and above at least one (remaining)
candidate contributes a whole vote to X's pairwise score versus Y and
vice versa,
a ballot that ranks X above Y contributes a whole vote to X's pairwise
score versus Y and nothing to Y's pairwise score
versus X.
Give each candidate X a score equal to X's smallest losing pairwise score.
Initially order the candidates from highest-scored to lowest scored. If
any adjacent pair is out-of-order pairwise, then swap
the out-of-order pair with the smallest score-difference. If there is a
tie for that then swap the tied pair that is lowest in
the order. Repeat until no adjacent pair is pairwise out-of-order, and
then eliminate the lowest-ordered candidate.
Repeat (disregarding any pairwise scores with eliminated candidates)
until one candidate remains. *
Some examples:
46 A>B
44 B>C (sincere is B or B>A)
05 C>A
05 C>B
A>B 51-49, B>C 90-10, C>A 54-46.
MinLV(erw) scores: B49 > A46 > C10.
Both adjacent pairs (B>A and A>C) are pairwise out of order. The B>A
score-difference is the smallest of the two
(3 versus 36) so we first swap that order to give
A49 > B51 > C10
Now neither pair of adjacent candidates is pairwise out of order so C is
eliminated and A wins.
Winning Votes, Margins, MMPO elect the Burier's candidate.
25 A>B
26 B>C
23 C>A
26 C
C>A 75-25, A>B 48-26, B>C 51-49.
MinLV(erw) scores: C49 > B26 > A25.
Both adjacent pairs (C>B and B>A) are pairwise out-of-order. The B-A
score difference is by
far the smallest, so we swap the B>A order to give
C > A > B. That order is final and C wins. C is the most top ranked
and the most above-bottom ranked
candidate. WV, MMPO, IRV, Benham elect B.
35 A
10 A=B
30 B>C
25 C
C>A 55-45, A>B 45-40 (note 10A=B effect), B>C 40-25.
MinLV(erw) scores: A45 > B40 > C25. Neither adjacent pair is pairwise
out-of-order so the order is final
and A wins.
A both pairwise-beats and positionally dominates B, but WV, Margins,
MMPO all elect B.
Chris Benham
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