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<p><br>
I made a strange mistake in the working out of my first example
(but not the result). Below it is corrected.<br>
<br>
(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<br>
was mostly copied from the initial uncorrected post. )<br>
<br>
Chris Benham<br>
</p>
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<br>
<br>
My favourite method that meets both Condorcet and Chicken Dilemma
is 'Min Losing-Votes (equal-ranking whole) Sorted Margins
Elimination':
<br>
<br>
*Voters rank from the top whatever number of candidates they like.
Equal-ranking and truncation are allowed.
<br>
<br>
For the purpose of determining candidates' pairwise scores:
<br>
<br>
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,
<br>
<br>
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,
<br>
<br>
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
<br>
versus X.
<br>
<br>
Give each candidate X a score equal to X's smallest losing
pairwise score.
<br>
<br>
Initially order the candidates from highest-scored to lowest
scored. If any adjacent pair is out-of-order pairwise, then swap
<br>
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
<br>
the order. Repeat until no adjacent pair is pairwise out-of-order,
and then eliminate the lowest-ordered candidate.
<br>
<br>
Repeat (disregarding any pairwise scores with eliminated
candidates) until one candidate remains. *
<br>
<br>
Some examples:
<br>
<br>
46 A>B
<br>
44 B>C (sincere is B or B>A)
<br>
05 C>A
<br>
05 C>B
<br>
<br>
A>B 51-49, B>C 90-10, C>A 54-46.
<br>
<br>
MinLV(erw) scores: B49 > A46 > C10.
<br>
<br>
Both adjacent pairs (B>A and A>C) are pairwise out of order.
The B>A score-difference is the smallest of the two<br>
(3 versus 36) so we first swap that order to give<br>
<br>
A49 > B51 > C10<br>
<br>
Now neither pair of adjacent candidates is pairwise out of order
so C is eliminated and A wins. <br>
<br>
Winning Votes, Margins, MMPO elect the Burier's candidate.<br>
<br>
25 A>B
<br>
26 B>C
<br>
23 C>A
<br>
26 C
<br>
<br>
C>A 75-25, A>B 48-26, B>C 51-49.
<br>
<br>
MinLV(erw) scores: C49 > B26 > A25.
<br>
<br>
Both adjacent pairs (C>B and B>A) are pairwise out-of-order.
The B-A score difference is by
<br>
far the smallest, so we swap the B>A order to give
<br>
<br>
C > A > B. That order is final and C wins. C is the most
top ranked and the most above-bottom ranked
<br>
candidate. WV, MMPO, IRV, Benham elect B.
<br>
<br>
35 A
<br>
10 A=B
<br>
30 B>C
<br>
25 C
<br>
<br>
C>A 55-45, A>B 45-40 (note 10A=B effect), B>C
40-25.
<br>
<br>
MinLV(erw) scores: A45 > B40 > C25. Neither adjacent pair
is pairwise out-of-order so the order is final
<br>
and A wins.
<br>
<br>
A both pairwise-beats and positionally dominates B, but WV,
Margins, MMPO all elect B.
<br>
<br>
Chris Benham
<br>
<br>
<br>
<br>
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