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<p class="MsoNormal">When beat path produces a tie, this method can produce a single winner unless the tie is genuine. It is the same method I presented earlier except for the addition of the Removing step, which resolves the ties. <br></p><p class="MsoNormal"><br></p><p class="MsoNormal">Candidates are classed in two categories: Winners and
Losers. Initially, all candidates are Winners. Every candidate has
an associated Set of candidates that includes itself and those candidates that
have defeated it. Every candidate initially has a set composed of itself
and no other candidates. Winners are those candidates who have no Winners
in their set aside from themselves.<br>
<br>
The pairs are ranked in order.<span style="mso-spacerun:yes"> </span>All pairs
are ranked in the form A>B indicating more voters rank A above B than rank B
above A.<span style="mso-spacerun:yes"> </span>Pairs with equal votes for A
above B and B above A are not ranked.<span style="mso-spacerun:yes"> </span>For
winning votes ranking, A>B is ranked higher than C>D if more voters
ranked A above B than ranked C above D.<span style="mso-spacerun:yes">
</span>If the same number of voters ranked A above B as ranked C above D then
A>B is ranked higher than C>D if more voters ranked D above C than ranked
B above A.<span style="mso-spacerun:yes"> </span>If the same number of voters
ranked A above B as ranked C above D and the same number ranked D above C as
ranked B above A then these pairs are equally ranked.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Affirm each group of equally ranked pairs in order, from
highest to lowest. <span style="mso-spacerun:yes"> </span>The count can
be ended before all pairs have been affirmed if only one Winner remains.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Affirming is composed of three steps: Combining sets,
Removing candidates from sets, and Reclassifying candidates.<span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Affirming Step 1: Combining</p>
<p class="MsoNormal">When A > B is affirmed, the set for candidate A is added
to every set that includes candidate B (not just candidate B’s set).<span style="mso-spacerun:yes"> </span>The Combining step is performed for all pairs
of the same rank before moving on to the Removing step.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Affirming Step 2: Removing</p>
<p class="MsoNormal">Each pair of winning candidates that are in each others'
sets are deleted from those sets. Example if C is in D's set and D is in C's
set and both C and D are winners, then delete C from D's set and delete D from
C's set.).<span style="mso-spacerun:yes"> </span>All such pairs of candidates
are removed before moving on to the Reclassifying step.<span style="mso-spacerun:yes"> </span>The inclusion of this step resolves ties that
are not resolved by the beat path method.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Affirming Step 3: Reclassifying</p>
<p class="MsoNormal">All Winners that now have Winning candidates other than
themselves in their set are reclassified as Losers.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Example</p>
<p class="MsoNormal">7 A > B > C > D<br>
6 C > D<br>
5 D > B > A > C<br style="mso-special-character:line-break">
<br style="mso-special-character:line-break">
</p>
<p class="MsoNormal" style="text-indent:.5in">A<span style="mso-tab-count:1"> </span>B<span style="mso-tab-count:1"> </span>C<span style="mso-tab-count:1"> </span>D<br>
A<span style="mso-tab-count:1"> </span>0 <span style="mso-tab-count:
1"> </span>7<span style="mso-tab-count:1"> </span>12<span style="mso-tab-count:1"> </span>7<br>
B<span style="mso-tab-count:1"> </span>5<span style="mso-tab-count:
1"> </span>0<span style="mso-tab-count:1"> </span>12<span style="mso-tab-count:1"> </span>7<br>
C<span style="mso-tab-count:1"> </span>6<span style="mso-tab-count:
1"> </span>6<span style="mso-tab-count:1"> </span>0<span style="mso-tab-count:1"> </span>13<br>
D<span style="mso-tab-count:1"> </span>11<span style="mso-tab-count:
1"> </span>11<span style="mso-tab-count:1"> </span>5<span style="mso-tab-count:1"> </span>0<br>
<br>
C> D 13,5<br>
A>C and B>C 12,6<br>
D>A and D>B 11, 7<br>
A>B 7,5<br>
<br>
affirm C>D<br>
A(W): A(W)<br>
B(W): B(W)<br>
C(W): C(W)<br>
D(L):C(W), D(L)<br>
D was reclassified as a Loser since C(W) is in its set. <br>
<br>
affirm A>C and B > C<br>
A(W): A(W)<br>
B(W): B(W)<br>
C(L): A(W), B(W), C(L)<br>
D(L): A(W), B(W), C(L), D(L)<br>
C was reclassified as a Loser since A(W) and B(W) are in its set. <br>
<br>
affirm D > A and D > B<br>
A(W): A(W), B(W)*, C(L), D(L)<br>
B(W): A(W)*, B(W), C(L), D(L)<br>
C(L): A(W), B(W), C(L), D(L)<br>
D(L): A(W), B(W), C(L), D(L)<br>
A and B are both winners. A is in B's set and B is in A's set. So A is deleted
from B's set and B is deleted from A's set.<br>
A(W): A(W), C(L), D(L)<br>
B(W): B(W), C(L), D(L)<br>
C(L): A(W), B(W), C(L), D(L)<br>
D(L): A(W), B(W), C(L), D(L)<br>
<br>
affirm A > B<br>
A(W):A(W), C(L), D(L)<br>
B(L): A(W), B(L), C(L), D(L)<br>
C(L): A(W), B(L), C(L), D(L)<br>
D(L): A(W), B(L), C(L), D(L)<br>
B was reclassified as a Loser since A(W) is in its set. <br style="mso-special-character:
line-break">
<br style="mso-special-character:line-break">
</p>
<p class="MsoNormal">A wins.<span style="mso-spacerun:yes"> </span>With beat
path, A and B are tied.<span style="mso-spacerun:yes"></span><br></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
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