[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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