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. 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

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 > > > > > ---- > Election-Methods mailing list - see http://electorama.com/em for list > info > > > ----- > No virus found in this message. > Checked by AVG - www.avg.com > Version: 2016.0.7797 / Virus Database: 4664/13132 - Release Date: > 10/02/16 > >