[EM] MinLV(erw) Sorted Margins Elimination (mistake in first example fixed)

[C.Benham]

Oops!   I made a blunder in my first example.   I've fixed it up below.

Chris Benham


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 
> 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 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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