<table cellspacing="0" cellpadding="0" border="0" ><tr><td valign="top" style="font: inherit;">I think it produces the Schultze result in fewer steps than Shultze if one is only interested in a single winner instead of the entire societal ordering. A societal ordering can also be produced.<!--[if gte mso 9]><xml>
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<![endif]--> In the example below, the second place winner can be found by rerunning the count,
this time beginning with Abby classed as a Loser and the other candidates
classed as Winners.<span style="mso-spacerun:yes"> </span>The second place
winner is Brad.<span style="mso-spacerun:yes"> </span>The third place winner
can be found by rerunning the count with Abby and Brad as Losers and the
remaining candidates as Winners. And so on.<span style="mso-spacerun:yes">
</span>The societal ordering produced by this method is Abby > Brad > Erin > Dave > Cora, the same as the Schulze
ordering.
<p class="MsoNormal"> </p>
<br><br><br> <br>--- On <b>Tue, 11/1/11, Jameson Quinn <i><jameson.quinn@gmail.com></i></b> wrote:<br><blockquote style="border-left: 2px solid rgb(16, 16, 255); margin-left: 5px; padding-left: 5px;"><br>From: Jameson Quinn <jameson.quinn@gmail.com><br>Subject: Re: [EM] non-transitive ranked pairs<br>To: "Ross Hyman" <rahyman@sbcglobal.net><br>Cc: election-methods@lists.electorama.com<br>Date: Tuesday, November 1, 2011, 7:37 AM<br><br><div id="yiv2097681541">At a first glance, this seems to be a definition that's equivalent to Schulze. <br><br><div class="yiv2097681541gmail_quote">2011/11/1 Ross Hyman <span dir="ltr"><<a rel="nofollow" ymailto="mailto:rahyman@sbcglobal.net" target="_blank" href="/mc/compose?to=rahyman@sbcglobal.net">rahyman@sbcglobal.net</a>></span><br>
<blockquote class="yiv2097681541gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"><table border="0" cellpadding="0" cellspacing="0"><tbody><tr><td style="font:inherit;" valign="top">
<p class="yiv2097681541MsoNormal">Ranked Pairs and all of its variants that I am aware of abhor
non-transitivity.<span> </span>Here is a variant of
Ranked Pairs that embraces non-transitivity.<span>
</span>Despite being non-transitive, it elects a unique winner, the Condorcet
winner if there is one.<span> </span>In the cases I
have looked at, the winner is also the Schulze winner.<span> </span>Is it always?</p>
<p class="yiv2097681541MsoNormal"> </p>
<p class="yiv2097681541MsoNormal">Candidates are classed in two categories: Winners and Losers.<span> </span>Initially, all candidates are Winners.<span> </span>Every candidate has an associated List of
candidates that have defeated it.<span> </span>Every
candidate initially has a List composed of itself and no other candidates.<span> </span>The method is so affirming of
non-transitivity that it even treats each candidate as a non-transitivity loop
unto itself.<span> </span>Winners are those
candidates who have no Winners in their List aside from themselves. </p>
<p class="yiv2097681541MsoNormal"> </p>
<p class="yiv2097681541MsoNormal">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> </span>Affirm each pair in order, from highest
ranked to lowest.<span> </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> </span>All
Winners that now have other Winners in their List are reclassified as
Losers.<span> </span>The count can be ended when only
one Winner remains since affirming the remaining pairs cannot make the Winner a
Loser.<span> </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="yiv2097681541MsoNormal"> </p>
<p class="yiv2097681541MsoNormal">Example election from:
<a rel="nofollow" target="_blank" href="http://www.cs.wustl.edu/%7Elegrand/rbvote/desc.html">http://www.cs.wustl.edu/~legrand/rbvote/desc.html</a> </p>
<p class="yiv2097681541MsoNormal"><span>Brad > Erin
623, 298</span></p>
<p class="yiv2097681541MsoNormal"><span>Erin</span><span> > Dave 610, 311</span></p>
<p class="yiv2097681541MsoNormal"><span>Dave > Brad 609, 312</span></p>
<p class="yiv2097681541MsoNormal"><span>Abby > Erin
511, 410</span></p>
<p class="yiv2097681541MsoNormal"><span>Abby > Dave 485, 436</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad > Abby 463, 458</span></p>
<p class="yiv2097681541MsoNormal"><span>Abby > Cora 461, 460</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad > Cora 461, 460</span></p>
<p class="yiv2097681541MsoNormal"><span>Dave > Cora 461, 460</span></p>
<p class="yiv2097681541MsoNormal"><span>Erin</span><span> > Cora 461, 460</span></p>
<p class="yiv2097681541MsoNormal"><span> </span></p>
<p class="yiv2097681541MsoNormal"><span>Each Candidate begins as a Winner with only
itself on its List.</span></p>
<p class="yiv2097681541MsoNormal"><span>Abby(W): Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad(W):Brad(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Cora(W): Cora(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Dave(W): Dave(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Erin(W): Erin(W)</span></p>
<p class="yiv2097681541MsoNormal"><span> </span></p>
<p class="yiv2097681541MsoNormal"><span>The first affirmed pair is Brad> Erin.<span> </span>Brad’s List is
added to Erin’s List, the only one that includes Erin.</span></p>
<p class="yiv2097681541MsoNormal"><span>Abby(W): Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad(W):Brad(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Cora(W): Cora(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Dave(W): Dave(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Erin(L): Erin(L), Brad(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Erin</span><span> is now a Looser.</span></p>
<p class="yiv2097681541MsoNormal"><span> </span></p>
<p class="yiv2097681541MsoNormal"><span>The next pair to be affirmed is Erin > Dave.<span> </span>Erin’s List is added to Dave’s List, the only one that
includes Dave.</span></p>
<p class="yiv2097681541MsoNormal"><span>Abby(W): Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad(W):Brad(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Cora(W): Cora(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Dave(L): Dave(L), Erin(L), Brad(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Erin(L): Erin(L), Brad(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Dave is now a Looser.</span></p>
<p class="yiv2097681541MsoNormal"><span> </span></p>
<p class="yiv2097681541MsoNormal"><span>The next pair to be affirmed is Dave >
Brad.<span> </span>Dave’s List is added to Brad’s and
Erin’s Lists, since both include Brad.</span></p>
<p class="yiv2097681541MsoNormal"><span>Abby(W): Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad(W):Brad(W), Dave(L), Erin(L)</span></p>
<p class="yiv2097681541MsoNormal"><span>Cora(W): Cora(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Dave(L): Dave(L), Erin(L), Brad(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Erin(L): Erin(L), Brad(W), Dave(L)</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad is still a Winner.</span></p>
<p class="yiv2097681541MsoNormal"><span> </span></p>
<p class="yiv2097681541MsoNormal"><span>The next pair to be affirmed is Abby > Erin.<span> </span>Abby’s List
is added to Brad’s, Dave’s, and Erin’s Lists, since they all include Erin.<span> </span></span></p>
<p class="yiv2097681541MsoNormal"><span>Abby(W): Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad(L):Brad(L), Dave(L), Erin(L), Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Cora(W): Cora(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Dave(L): Dave(L), Erin(L), Brad(L), Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Erin(L): Erin(L), Brad(L), Dave(L), Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad is now a Looser.</span></p>
<p class="yiv2097681541MsoNormal"> </p>
<p class="yiv2097681541MsoNormal">The next pair to be affirmed is Abby > Dave.<span> </span>The Lists do not change.</p>
<p class="yiv2097681541MsoNormal"> </p>
<p class="yiv2097681541MsoNormal">The next pair to be affirmed is Brad > Abby.<span> </span></p>
<p class="yiv2097681541MsoNormal"><span>Abby(W): Abby(W), Brad(L), Dave(L), Erin(L)</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad(L):Brad(L), Dave(L), Erin(L), Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Cora(W): Cora(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Dave(L): Dave(L), Erin(L), Brad(L), Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Erin(L): Erin(L), Brad(L), Abby(W), Dave(L)</span></p>
<p class="yiv2097681541MsoNormal">Abby is still a Winner.</p>
<p class="yiv2097681541MsoNormal"> </p>
<p class="yiv2097681541MsoNormal">The next affirmed pair is Abby > Cora.<span> </span></p>
<p class="yiv2097681541MsoNormal"><span>Abby(W): Abby(W), Brad(L), Dave(L), Erin(L)</span></p>
<p class="yiv2097681541MsoNormal"><span>Brad(L):Brad(L), Dave(L), Erin(L), Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Cora(L): Cora(L), Abby(W), Brad(L),
Dave(L), Erin(L)</span></p>
<p class="yiv2097681541MsoNormal"><span>Dave(L): Dave(L), Erin(L), Brad(L), Abby(W)</span></p>
<p class="yiv2097681541MsoNormal"><span>Erin(L): Erin(L), Brad(L), Abby(W), Dave(L)</span></p>
<p class="yiv2097681541MsoNormal">Cora is now a looser.</p>
<p class="yiv2097681541MsoNormal"> </p>
<p class="yiv2097681541MsoNormal">Abby is the winner of the election.</p>
</td></tr></tbody></table><br>----<br>
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<br></blockquote></div><br>
</div></blockquote></td></tr></table>