[EM] MinLV(erw) Sorted Margins Elimination (mistake in first example fixed)
cbenham at adam.com.au
Tue Oct 4 06:13:49 PDT 2016
Oops! I made a blunder in my first example. I've fixed it up below.
On 10/3/2016 2:35 AM, C.Benham wrote:
> My favourite method that meets both Condorcet and Chicken Dilemma is
> 'MinLosing 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 truncates both X and Y contributes nothing to X's
> pairwise score versus Y and vice versa,
> a ballot that ranks X and Y equal (above bottom) 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
> 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 3 candidates remain and then elect the
> highest-ordered candidate.*
> (Using the number "3" at the end instead of 1 is just a time-saver.)
> The part of the algorithm that combines candidates' scores with
> pairwise results to order the candidates (as used in Approval
> Sorted Margins) is an excellent invention of Forest Simmons.
> It doesn't meet Unburiable Mutual Dominant Third, which means that it
> doesn't dominate Benham.
> (That criterion says that if the winner X is part of a set S of
> candidates who are ranked above all outside-S candidates on
> more than a third of the ballots, and all candidates in S pairwise
> beat all outside-S candidates, then it isn't possible to change
> some ballots that rank some outside-S Y above X so with the effect of
> changing the winner from Y to X.)
> This meets Smith, Plurality, Mono-raise, Mono-switch-plump,
> Non-drastic Defense.
> If candidate A is pairwise-beaten by B and positionally dominated by B
> then B can't win.
> If there is a positionally dominant and uncovered X, then I claim X
> will win.
> Minimal Defense is incompatible with Chicken Dilemma, and FBC is
> incompatible with Condorcet.
> 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
> difference in scores is by far the smallest between A and B, so
> we switch that order to give A > B > C. Now neither adjacent pair
> (A>B or B>C) is pairwise out-of-order, so that order is final 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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