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<p class="MsoNormal">Ranked Pairs and all of its variants that I am aware of abhor
non-transitivity.<span style="mso-spacerun:yes"> </span>Here is a variant of
Ranked Pairs that embraces non-transitivity.<span style="mso-spacerun:yes">
</span>Despite being non-transitive, it elects a unique winner, the Condorcet
winner if there is one.<span style="mso-spacerun:yes"> </span>In the cases I
have looked at, the winner is also the Schulze winner.<span style="mso-spacerun:yes"> </span>Is it always?</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Candidates are classed in two categories: Winners and Losers.<span style="mso-spacerun:yes"> </span>Initially, all candidates are Winners.<span style="mso-spacerun:yes"> </span>Every candidate has an associated List of
candidates that have defeated it.<span style="mso-spacerun:yes"> </span>Every
candidate initially has a List composed of itself and no other candidates.<span style="mso-spacerun:yes"> </span>The method is so affirming of
non-transitivity that it even treats each candidate as a non-transitivity loop
unto itself.<span style="mso-spacerun:yes"> </span>Winners are those
candidates who have no Winners in their List aside from themselves. </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Rank the pairs in a strict order, in the same order one
would use for your favorite strict order transitive variant of Ranked Pairs.<span style="mso-spacerun:yes"> </span>Affirm each pair in order, from highest
ranked to lowest.<span style="mso-spacerun:yes"> </span>When A > B is
affirmed, the List for candidate A is added to every List that includes candidate
B (not just candidate B’s list).<span style="mso-spacerun:yes"> </span>All
Winners that now have other Winners in their List are reclassified as
Losers.<span style="mso-spacerun:yes"> </span>The count can be ended when only
one Winner remains since affirming the remaining pairs cannot make the Winner a
Loser.<span style="mso-spacerun:yes"> </span>Provided that every pair is
ranked into a strict ranking, and each pair expresses a definite ranking
between the two candidates in the pair, there is guaranteed to be one Winner.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Example election from:
http://www.cs.wustl.edu/~legrand/rbvote/desc.html </p>
<p class="MsoNormal"><span class="cand">Brad > Erin
623, 298</span></p>
<p class="MsoNormal"><span class="cand">Erin</span><span class="cand"> > Dave 610, 311</span></p>
<p class="MsoNormal"><span class="cand">Dave > Brad 609, 312</span></p>
<p class="MsoNormal"><span class="cand">Abby > Erin
511, 410</span></p>
<p class="MsoNormal"><span class="cand">Abby > Dave 485, 436</span></p>
<p class="MsoNormal"><span class="cand">Brad > Abby 463, 458</span></p>
<p class="MsoNormal"><span class="cand">Abby > Cora 461, 460</span></p>
<p class="MsoNormal"><span class="cand">Brad > Cora 461, 460</span></p>
<p class="MsoNormal"><span class="cand">Dave > Cora 461, 460</span></p>
<p class="MsoNormal"><span class="cand">Erin</span><span class="cand"> > Cora 461, 460</span></p>
<p class="MsoNormal"><span class="cand"> </span></p>
<p class="MsoNormal"><span class="cand">Each Candidate begins as a Winner with only
itself on its List.</span></p>
<p class="MsoNormal"><span class="cand">Abby(W): Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Brad(W):Brad(W)</span></p>
<p class="MsoNormal"><span class="cand">Cora(W): Cora(W)</span></p>
<p class="MsoNormal"><span class="cand">Dave(W): Dave(W)</span></p>
<p class="MsoNormal"><span class="cand">Erin(W): Erin(W)</span></p>
<p class="MsoNormal"><span class="cand"> </span></p>
<p class="MsoNormal"><span class="cand">The first affirmed pair is Brad> Erin.<span style="mso-spacerun:yes"> </span>Brad’s List is
added to Erin’s List, the only one that includes Erin.</span></p>
<p class="MsoNormal"><span class="cand">Abby(W): Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Brad(W):Brad(W)</span></p>
<p class="MsoNormal"><span class="cand">Cora(W): Cora(W)</span></p>
<p class="MsoNormal"><span class="cand">Dave(W): Dave(W)</span></p>
<p class="MsoNormal"><span class="cand">Erin(L): Erin(L), Brad(W)</span></p>
<p class="MsoNormal"><span class="cand">Erin</span><span class="cand"> is now a Looser.</span></p>
<p class="MsoNormal"><span class="cand"> </span></p>
<p class="MsoNormal"><span class="cand">The next pair to be affirmed is Erin > Dave.<span style="mso-spacerun:yes"> </span>Erin’s List is added to Dave’s List, the only one that
includes Dave.</span></p>
<p class="MsoNormal"><span class="cand">Abby(W): Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Brad(W):Brad(W)</span></p>
<p class="MsoNormal"><span class="cand">Cora(W): Cora(W)</span></p>
<p class="MsoNormal"><span class="cand">Dave(L): Dave(L), Erin(L), Brad(W)</span></p>
<p class="MsoNormal"><span class="cand">Erin(L): Erin(L), Brad(W)</span></p>
<p class="MsoNormal"><span class="cand">Dave is now a Looser.</span></p>
<p class="MsoNormal"><span class="cand"> </span></p>
<p class="MsoNormal"><span class="cand">The next pair to be affirmed is Dave >
Brad.<span style="mso-spacerun:yes"> </span>Dave’s List is added to Brad’s and
Erin’s Lists, since both include Brad.</span></p>
<p class="MsoNormal"><span class="cand">Abby(W): Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Brad(W):Brad(W), Dave(L), Erin(L)</span></p>
<p class="MsoNormal"><span class="cand">Cora(W): Cora(W)</span></p>
<p class="MsoNormal"><span class="cand">Dave(L): Dave(L), Erin(L), Brad(W)</span></p>
<p class="MsoNormal"><span class="cand">Erin(L): Erin(L), Brad(W), Dave(L)</span></p>
<p class="MsoNormal"><span class="cand">Brad is still a Winner.</span></p>
<p class="MsoNormal"><span class="cand"> </span></p>
<p class="MsoNormal"><span class="cand">The next pair to be affirmed is Abby > Erin.<span style="mso-spacerun:yes"> </span>Abby’s List
is added to Brad’s, Dave’s, and Erin’s Lists, since they all include Erin.<span style="mso-spacerun:yes"> </span></span></p>
<p class="MsoNormal"><span class="cand">Abby(W): Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Brad(L):Brad(L), Dave(L), Erin(L), Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Cora(W): Cora(W)</span></p>
<p class="MsoNormal"><span class="cand">Dave(L): Dave(L), Erin(L), Brad(L), Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Erin(L): Erin(L), Brad(L), Dave(L), Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Brad is now a Looser.</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The next pair to be affirmed is Abby > Dave.<span style="mso-spacerun:yes"> </span>The Lists do not change.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The next pair to be affirmed is Brad > Abby.<span style="mso-spacerun:yes"> </span></p>
<p class="MsoNormal"><span class="cand">Abby(W): Abby(W), Brad(L), Dave(L), Erin(L)</span></p>
<p class="MsoNormal"><span class="cand">Brad(L):Brad(L), Dave(L), Erin(L), Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Cora(W): Cora(W)</span></p>
<p class="MsoNormal"><span class="cand">Dave(L): Dave(L), Erin(L), Brad(L), Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Erin(L): Erin(L), Brad(L), Abby(W), Dave(L)</span></p>
<p class="MsoNormal">Abby is still a Winner.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The next affirmed pair is Abby > Cora.<span style="mso-spacerun:yes"> </span></p>
<p class="MsoNormal"><span class="cand">Abby(W): Abby(W), Brad(L), Dave(L), Erin(L)</span></p>
<p class="MsoNormal"><span class="cand">Brad(L):Brad(L), Dave(L), Erin(L), Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Cora(L): Cora(L), Abby(W), Brad(L),
Dave(L), Erin(L)</span></p>
<p class="MsoNormal"><span class="cand">Dave(L): Dave(L), Erin(L), Brad(L), Abby(W)</span></p>
<p class="MsoNormal"><span class="cand">Erin(L): Erin(L), Brad(L), Abby(W), Dave(L)</span></p>
<p class="MsoNormal">Cora is now a looser.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Abby is the winner of the election.</p>
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