<div dir="ltr"><div dir="ltr"><p class="MsoNormal">Dear gentlemen, <span></span></p><p class="MsoNormal"><br></p>
<p class="MsoNormal">A while ago I did write here about the <a href="https://electowiki.org/wiki/Iterated_Bucklin">Iterated Bucklin</a>
method on which I’ve recently had a chance to think and generalize about a bit
more. Maybe some of the below could be novel or otherwise of interest.<span></span></p><p class="MsoNormal"><br></p>
<p class="MsoNormal">First, and for our purposes today, let's define the <b>Serious
Candidates Set</b> in the context of a ranked ballot, to include those candidates
who would win an approval count if they served as the approval cutoff across
all ballots.<span></span></p><p class="MsoNormal"><br></p>
<p class="MsoNormal">In the [2:A>B, 3:C>A, 4:A>B] election as an
example, the Set would include A and B only, as applying the cutoff at C would
still elect B.<span></span></p><p class="MsoNormal"><br></p>
<p class="MsoNormal">I’ve been checking some random simulations from Kevin
Venzke’s <a href="http://votingmethods.net">votingmethods.net</a>, and here are some properties of this Set that I
*suspect*:<span></span></p>
<p class="gmail-MsoListParagraphCxSpFirst" style="">1.<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-size-adjust:none;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span>If there is a Condorcet Winner, this Set should
always include them. <span></span></p>
<p class="gmail-MsoListParagraphCxSpLast" style="">2.<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-size-adjust:none;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span>Otherwise, this Set should always partially
overlap with the Smith Set.<span></span></p>
<p class="MsoNormal">Now, quite a few methods emerge once the Serious Candidate
Set is isolated (by actually checking the approval winner once every candidate
is used as a cutoff). The five below allow truncation and equal ranking, and
have been checked (again courtesy of <a href="http://votingmethods.net">votingmethods.net</a>) to ensure that they are
different from one-another and the 40-odd other methods Kevin has aggregated
over there. <span></span></p><p class="MsoNormal"><br></p>
<p class="MsoNormal">So, which member of the Serious Candidate Set should be
elected?<span></span></p>
<p class="gmail-MsoListParagraphCxSpFirst" style="">1.<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-size-adjust:none;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span>Electing the Serious Candidate that wins their
cutoff count by the most approvals. Rather obvious but not too much of an
improvement over Approval (if any at all). Terrible Later No Harm failures,
though this is in the context where truncation is allowed. Fails Condorcet.<span></span></p>
<p class="gmail-MsoListParagraphCxSpMiddle" style="">2.<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-size-adjust:none;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span>Electing the Serious Candidate that wins their
cutoff count by the *least* approvals. A bit counterintuitive, but winning by
the least means that the winner had to “dip” the least into each approver’s
rankings. If this is not compliant with Later No Harm, it should at least fail
rarely. It would fail Later No Help spectacularly though, indeed having a huge
incentive to always rank your least favorite candidate that is still likely to
win last, instead of leaving them unranked. Unfortunately, I’ve seen it elect
the Condorcet Loser at least once. <span></span></p>
<p class="gmail-MsoListParagraphCxSpMiddle" style="">3.<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-size-adjust:none;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span>Electing the Serious Candidate that wins their
cutoff count by the most approvals *compared to the runner up*. May fail Condorcet the least. Likely the most sensible of the bunch. <span></span></p>
<p class="gmail-MsoListParagraphCxSpMiddle" style="">4.<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-size-adjust:none;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span>Iterated Bucklin (now fitting into this generalised family) will always elect a member of
the Set, but it seems to be neither of the three above with consistency. I
cannot seem to find the pattern the method lands on.<span></span></p>
<p class="gmail-MsoListParagraphCxSpLast" style="">5.<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-variant-alternates:normal;font-size-adjust:none;font-kerning:auto;font-feature-settings:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span>Electing the Serious Candidate that wins the
election if the cutoff is set at the FPP winner. If the FPP winner is in the
Set to begin with, they will be elected. Otherwise, again a method that elects
a winner from the set through no obvious pattern. Of particular interest to me
since it’s the only method in here that can be hand-counted with relative ease
(it’s just an FPP count and an approval count after that).</p><p class="gmail-MsoListParagraphCxSpLast" style="">For reference, standard Bucklin may not always elect members of the Set so cannot be retconned into this tree. I've tested quite a few other methods, and there are some for which I'm still to find a failure to elect from the Serious Candidates Set, including Borda and, unsurprisingly, many approval variations and Approval-Condorcet hybrids. </p><p class="gmail-MsoListParagraphCxSpLast" style="">Just some preliminary thoughts
above, hopefully of some interest.</p><p class="gmail-MsoListParagraphCxSpLast" style=""><br></p><p class="gmail-MsoListParagraphCxSpLast" style="">Best regards,</p><p class="gmail-MsoListParagraphCxSpLast" style="">Etjon Basha</p></div></div>