<div dir="auto"><div>I wonder what Max Strength Minimal Covering Chain does with these ballot profiles.<div dir="auto"><br></div><div dir="auto">A chain is a beatpath in which each alternative defeats every subsequent alternative, not only the immediately subsequent alternative.</div><div dir="auto"><br></div><div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">[The strength of a chain is the defeat strength of the head>tail defeat.]</div><div dir="auto"><br></div><div dir="auto">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.</div><br><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Apr 30, 2023, 1:05 PM C.Benham <<a href="mailto:cbenham@adam.com.au" target="_blank" rel="noreferrer">cbenham@adam.com.au</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">(I first suggested this method here in October 2016, but with a blunder <br>
in the last paragraph.)<br>
<br>
My favourite method that meets both Condorcet and Chicken Dilemma is <br>
'Min Losing-Votes (equal-ranking whole) Sorted Margins Elimination':<br>
<br>
*Voters rank from the top whatever number of candidates they like. <br>
Equal-ranking and truncation are allowed.<br>
<br>
For the purpose of determining candidates' pairwise scores:<br>
<br>
a ballot that votes both X and Y above no other (remaining) candidates <br>
contributes nothing to X's pairwise score versus Y and vice versa,<br>
<br>
a ballot that ranks X and Y equal and above at least one (remaining) <br>
candidate contributes a whole vote to X's pairwise score versus Y and <br>
vice versa,<br>
<br>
a ballot that ranks X above Y contributes a whole vote to X's pairwise <br>
score versus Y and nothing to Y's pairwise score<br>
versus X.<br>
<br>
Give each candidate X a score equal to X's smallest losing pairwise score.<br>
<br>
Initially order the candidates from highest-scored to lowest scored. If <br>
any adjacent pair is out-of-order pairwise, then swap<br>
the out-of-order pair with the smallest score-difference. If there is a <br>
tie for that then swap the tied pair that is lowest in<br>
the order. Repeat until no adjacent pair is pairwise out-of-order, and <br>
then eliminate the lowest-ordered candidate.<br>
<br>
Repeat (disregarding any pairwise scores with eliminated candidates) <br>
until one candidate remains. *<br>
<br>
Some examples:<br>
<br>
46 A>B<br>
44 B>C (sincere is B or B>A)<br>
05 C>A<br>
05 C>B<br>
<br>
A>B 51-49, B>C 90-10, C>A 54-46.<br></blockquote></div></div><div dir="auto"><br></div><div dir="auto">The respective WV+LA defeat strengths are</div><div dir="auto">A> B 51+5, B>C 90+51, and C>A 54+49</div><div dir="auto">The strongest is B>C ... the burier wins.</div><div dir="auto"><br></div><div dir="auto">The respective Implicit Approval Margins are 51-95, 95-54, and 54-51 ... again B wins.</div><div dir="auto"><br></div><div dir="auto">Let's try chain Climbing on the UTBO:</div><div dir="auto">A<C<B ... so A is eliminated, taking down B with it, leaving C.</div><div dir="auto"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<br>
MinLV(erw) scores: B49 > A46 > C10.<br>
<br>
Neither adjacent pair (B>A or A>C) is pairwise out of order, so that <br>
order is final, and as there are only 3 candidates then A wins.<br>
<br>
Winning Votes, Margins, MMPO elect the Burier's candidate.<br>
<br>
25 A>B<br>
26 B>C<br>
23 C>A<br>
26 C<br>
<br>
C>A 75-25, A>B 48-26, B>C 51-49.<br></blockquote></div></div><div dir="auto"><br></div><div dir="auto"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"></blockquote></div></div><div dir="auto">The respective WV+LA defeat strengths are A>B 48+49, B>C 51+25, and</div><div dir="auto">C>A 75+52 ... So C wins.</div><div dir="auto"><br></div><div dir="auto">How about chain climbing? The UTB Order is A(48)<B(51)<C(75) ... A takes down B with it, leaving C.</div><div dir="auto"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<br>
MinLV(erw) scores: C49 > B26 > A25.<br>
<br>
Both adjacent pairs (C>B and B>A) are pairwise out-of-order. The B-A <br>
score difference is by<br>
far the smallest, so we swap the B>A order to give<br>
<br>
C > A > B. That order is final and C wins. C is the most top ranked <br>
and the most above-bottom ranked<br>
candidate. WV, MMPO, IRV, Benham elect B.<br>
<br>
35 A<br>
10 A=B<br>
30 B>C<br>
25 C<br>
<br>
C>A 55-45, A>B 45-40 (note 10A=B effect), B>C 40-25.<br></blockquote></div></div><div dir="auto"><br></div><div dir="auto"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"></blockquote></div></div><div dir="auto">WV+LA scores are A>B 45+60, </div><div dir="auto">B>C 40+45, C>A 55+55 ... so C wins with A A not far behind.</div><div dir="auto"><br></div><div dir="auto">How a out chain climbing? The UTBOrder is B(40)<A(45)<C(55)</div><div dir="auto">Candidate B takes down C with it, leaving only A.</div><div dir="auto"><br></div><div dir="auto">It looks like Max Strength Minimal Covering Chain elects the burier only in the first example.</div><div dir="auto"><br></div><div dir="auto">Universal Tie Breaking Order Chain Climbing never elects the burier.</div><div dir="auto"><br></div><div dir="auto">It has the advantage of being monotonic and never electing a covered candidate ... but fails reverse symmetry.</div><div dir="auto"><br></div><div dir="auto">-fws</div><div dir="auto"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<br>
MinLV(erw) scores: A45 > B40 > C25. Neither adjacent pair is pairwise <br>
out-of-order so the order is final<br>
and A wins.<br>
<br>
A both pairwise-beats and positionally dominates B, but WV, Margins, <br>
MMPO all elect B.<br>
<br>
Chris Benham<br>
<br>
<br>
<br>
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</blockquote></div></div></div>