<div dir="ltr"><p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">I've
got a new fpA-fpC-compliant method and this time I am very satisfied with its
behaviour (after checking some test examples that some other methods failed, though
I guess it needs some robust checks on monotonicity). Instead of just pairwise
comparing the candidates, it compares the "alliances" of these
candidates in order to create a special pairwise comparison matrix (in a more
burial-resistant manner). The mechanism tends to break Condorcet cycles, even
though there are some rare cases where it doesn't clear <i>all</i> the ambiguity
and a completion method has to be used.</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">The
"agreement" rule was created specifically to make the method
cloneproof (but it also had to be strict enough so that a) the burial
resistance was not weakened, b) the method didn't stop converging, i.e. achieving
stable solutions).</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">It's
worth noting that - in a given iteration - a covered candidate can't have any
allies against the covering candidate, so a covering relation should cause a
visibly strong defeat.</span></p><p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif"><br></span></p><p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><b style="font-size:11pt"><span style="font-family:Arial,sans-serif">Convergent
Allied Comparison (CAC)</span></b><br></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">Definitions:</span></p>
<p class="gmail-MsoListParagraphCxSpFirst" style="text-align:justify;margin:0cm 0cm 0cm 36pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Symbol">·<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span><span style="font-family:Arial,sans-serif">(A>B)-ally
- a candidate that pairwise beats B, doesn't pairwise beat A and is not in
agreement with B against A</span></p>
<p class="gmail-MsoListParagraphCxSpMiddle" style="text-align:justify;margin:0cm 0cm 0cm 36pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Symbol">·<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span><span style="font-family:Arial,sans-serif">(A,X)-agreement
against B - occurs when A and X are in the same pairwise relation with B (win,
lose or tie), number of A>=X>B votes is greater than number of
A>=B>X votes and number of X>=A>B votes is greater than number of
X>=B>A votes</span></p>
<p class="gmail-MsoListParagraphCxSpMiddle" style="text-align:justify;margin:0cm 0cm 0cm 36pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Symbol">·<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span><span style="font-family:Arial,sans-serif">(A>B)-alliance
- a set that consists of A and all (A>B)-allies</span></p>
<p class="gmail-MsoListParagraphCxSpLast" style="text-align:justify;margin:0cm 0cm 8pt 36pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Symbol">·<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span><span style="font-family:Arial,sans-serif">score(A>B)
= [number of votes that prefer some member of the (A>B)-alliance to every
member of the (B>A)-alliance] + [number of votes that prefer A to every
member of the (B>A)-alliance if A pairwise beats B, otherwise 0]</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">Procedure:</span></p>
<p class="gmail-MsoListParagraphCxSpFirst" style="text-align:justify;margin:0cm 0cm 0cm 36pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Symbol">·<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span><span style="font-family:Arial,sans-serif">Calculate
the scores and use them to create a pairwise comparison matrix - then use this
matrix to redefine which candidates are pairwise beaten by whom (which in turn
can affect the alliances and the scores). Repeat until the comparison matrix
does not change with a new iteration - this is the final CAC matrix.</span></p><p class="gmail-MsoListParagraphCxSpFirst" style="text-align:justify;margin:0cm 0cm 0cm 36pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:11pt;font-family:Symbol">·<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span><span style="font-size:11pt;font-family:Arial,sans-serif">If
there exists a candidate that is unbeaten according to the CAC matrix, choose
this candidate. Otherwise apply a defeat-dropping Condorcet method (River,
RP/MAM or Schulze) to the CAC matrix.</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif;font-size:11pt"><br></span></p><p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif;font-size:11pt">Example:</span><br></p>
<p class="MsoNormal" style="margin:0cm;text-align:justify;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">9: A>B>C</span></p>
<p class="MsoNormal" style="margin:0cm;text-align:justify;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">6: B>C>A</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">8:
C>A>B</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">There's
no agreement between any candidates. (A>C)-alliance is {A,B}, (B>A)-alliance is {B,C},
and (C>B)-alliance is {A,C}. Scores in the first iteration are as follows:</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:"Courier New""> | A | B | C</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:"Courier New"">A
| -
| 9+9=18 | 9+6=15</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:"Courier New"">B
| 6+8=14 | - | 6+6=12</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:"Courier New"">C
| 8+8=16 | 9+8=17 | - </span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">So
we already see a strict order (C>A>B) in this matrix, but as the beating
relations (and, with them, the alliances and the scores) have changed, let's see the
next iteration to make sure that the solution is stable. Now the only multi-member
alliance is the (C>B)-alliance: {A,C}.</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:"Courier New""> | A | B | C</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:"Courier New"">A
| - | 9+8+9+8=34
| 9 </span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:"Courier New"">B
| 6 | - | 6</span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:"Courier New"">C
| 6+8+6+8=28 | 9+8+8=25 | - </span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">The
pairwise relations remain unchanged, so this is the final CAC matrix, with C
candidate confirmed as the winner.</span></p><p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif">- Filip Ejlak</span></p></div>