[EM] MinLV(erw) Sorted Margins Elimination

C.Benham cbenham at adam.com.au
Sun Oct 2 09:05:12 PDT 2016

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 

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 

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.

Neither adjacent pair (B>A or A>C) is pairwise out of order, so that 
order is final, and as there are only 3 candidates then 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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