(I first suggested this method here in October 2016, but with a blunder in the last paragraph.) My favourite method that meets both Condorcet and Chicken Dilemma is 'Min Losing-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 votes both X and Y above no other (remaining) candidates contributes nothing to X's pairwise score versus Y and vice versa, 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, 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 one candidate  remains. * 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

I wonder what Max Strength Minimal Covering Chain does with these ballot profiles. A chain is a beatpath in which each alternative defeats every subsequent alternative, not only the immediately subsequent alternative. It covers if each candidate that is not in the chain is defeated by some candidate that is in the chain. If no proper subset covers it is a minimal cover. In each of the examples below each of the three defeats constitutes a minimal covering chain (and there are no others). So we just focus on finding the strength of each defeat. [The strength of a chain is the defeat strength of the head>tail defeat.] Various gauges of defeat strength are possible. One I like is winning votes plus losing abstentions. Another is winner implicit approval minus loser implicit approval ... which takes advantage of the "Universal Tie Breaking Order" that extends implicit approval into a highly conclusive method. On Sun, Apr 30, 2023, 1:05 PM C.Benham <cbenham@adam.com.au> wrote: > (I first suggested this method here in October 2016, but with a blunder > in the last paragraph.) > > My favourite method that meets both Condorcet and Chicken Dilemma is > 'Min Losing-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 votes both X and Y above no other (remaining) candidates > contributes nothing to X's pairwise score versus Y and vice versa, > > 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, > > 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 one candidate remains. * > > 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. > The respective WV+LA defeat strengths are A> B 51+5, B>C 90+51, and C>A 54+49 The strongest is B>C ... the burier wins. The respective Implicit Approval Margins are 51-95, 95-54, and 54-51 ... again B wins. Let's try chain Climbing on the UTBO: A<C<B ... so A is eliminated, taking down B with it, leaving C. > > 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. > The respective WV+LA defeat strengths are A>B 48+49, B>C 51+25, and C>A 75+52 ... So C wins. How about chain climbing? The UTB Order is A(48)<B(51)<C(75) ... A takes down B with it, leaving C. > > 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. > WV+LA scores are A>B 45+60, B>C 40+45, C>A 55+55 ... so C wins with A A not far behind. How a out chain climbing? The UTBOrder is B(40)<A(45)<C(55) Candidate B takes down C with it, leaving only A. It looks like Max Strength Minimal Covering Chain elects the burier only in the first example. Universal Tie Breaking Order Chain Climbing never elects the burier. It has the advantage of being monotonic and never electing a covered candidate ... but fails reverse symmetry. -fws > > 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 https://electorama.com/em for list > info >