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</div><div id="yui_3_16_0_1_1450286497486_3426" class="">
<div id="yui_3_16_0_1_1450286497486_3428" class=""><span class="" id="yui_3_16_0_1_1450286497486_3430" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">Proof that the ranked Phragmen method (QPQ) satisfies
proportional completion.<span class="" id="yui_3_16_0_1_1450286497486_3432" style="mso-spacerun:yes"> </span></span></div>
<div id="yui_3_16_0_1_1450286497486_3434" class=""><span class="" id="yui_3_16_0_1_1450286497486_3436" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"><br></span></div><div id="yui_3_16_0_1_1450286497486_3669" class=""><span class="" id="yui_3_16_0_1_1450286497486_3436" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">The Phragmen Method<br></span></div><div class=""><span class="" id="yui_3_16_0_1_1450286497486_3436" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">The Phragmen method is, in my opinion, a beautifully
elegant way to elect multiple winners proportionally from Approval
ballots.<span class="" id="yui_3_16_0_1_1450286497486_3438" style="mso-spacerun:yes"> </span>It was introduced to this list
by Olli Salmi in 2002.<span class="" id="yui_3_16_0_1_1450286497486_3440" style="mso-spacerun:yes"> </span></span></div>
<div id="yui_3_16_0_1_1450286497486_3442" class=""><span class="" id="yui_3_16_0_1_1450286497486_3444" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"><a class="" id="yui_3_16_0_1_1450286497486_3446" href="http://groups.yahoo.com/group/election-methods-list/message/10068?soc_src=mail&soc_trk=ma">http://groups.yahoo.com/group/election-methods-list/message/10068</a></span></div>
<div id="yui_3_16_0_1_1450286497486_3674" class=""><span class="" id="yui_3_16_0_1_1450286497486_3450" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"></span><span class="" id="yui_3_16_0_1_1450286497486_3450" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">I think it would be the simplest and most elegant way
to modify the nomination procedure for the Hugo.</span></div><div id="yui_3_16_0_1_1450286497486_3686" class=""><span class="" id="yui_3_16_0_1_1450286497486_3450" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"><br></span></div>
<div id="yui_3_16_0_1_1450286497486_3452" class=""><span class="" id="yui_3_16_0_1_1450286497486_3454" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">Salmi generalized Phragmen to ranked ballots and the
method was developed further by Douglas Woodall in Voting Matters,
where he christened it QPQ.<span class="" id="yui_3_16_0_1_1450286497486_3456" style="mso-spacerun:yes"> </span></span></div>
<div id="yui_3_16_0_1_1450286497486_3458" class=""><span class="" id="yui_3_16_0_1_1450286497486_3460" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"><a class="" id="yui_3_16_0_1_1450286497486_3462" href="http://www.votingmatters.org.uk/ISSUE17/I17P1.PDF">http://www.votingmatters.org.uk/ISSUE17/I17P1.PDF</a></span></div>
<div id="yui_3_16_0_1_1450286497486_3687" class=""><span class="" id="yui_3_16_0_1_1450286497486_3466" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"></span><span class="" id="yui_3_16_0_1_1450286497486_3466" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">The method is, in my opinion, a very beautiful and
elegant STV-like method.<span class="" id="yui_3_16_0_1_1450286497486_3468" style="mso-spacerun:yes"> </span>I have also
written about it in Voting Matters.<span class="" id="yui_3_16_0_1_1450286497486_3470" style="mso-spacerun:yes"> </span></span></div>
<div id="yui_3_16_0_1_1450286497486_3472" class=""><span class="" id="yui_3_16_0_1_1450286497486_3474" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"><a class="" id="yui_3_16_0_1_1450286497486_3476" href="http://www.votingmatters.org.uk/ISSUE28/I28P2.pdf">http://www.votingmatters.org.uk/ISSUE28/I28P2.pdf</a><span class="" id="yui_3_16_0_1_1450286497486_3478" style="mso-field-code:" HYPERLINK \\h ""><span id="yui_3_16_0_1_1450286497486_3480" class=""> </span></span></span></div>
<div id="yui_3_16_0_1_1450286497486_3482" class=""><span class="" id="yui_3_16_0_1_1450286497486_3484" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"><br></span></div><div id="yui_3_16_0_1_1450286497486_3688" class=""><span class="" id="yui_3_16_0_1_1450286497486_3484" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">Proportional Completion<br></span></div><div class=""><span class="" id="yui_3_16_0_1_1450286497486_3484" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">Proportional completion is a way to complete ballots
that do not rank every candidate.<span class="" id="yui_3_16_0_1_1450286497486_3486" style="mso-spacerun:yes"> </span>It was
developed by Markus Schulze and used in his multiple winner proportional
representation version of his beat path voting method, Schulze STV.<span class="" id="yui_3_16_0_1_1450286497486_3488" style="mso-spacerun:yes"> </span></span></div>
<div id="yui_3_16_0_1_1450286497486_3490" class=""><span class="" id="yui_3_16_0_1_1450286497486_3492" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"><a class="" id="yui_3_16_0_1_1450286497486_3494" href="http://m-schulze.9mail.de/schulze2.pdf">http://m-schulze.9mail.de/schulze2.pdf</a></span></div>
<div id="yui_3_16_0_1_1450286497486_3496" class=""><span class="" id="yui_3_16_0_1_1450286497486_3498" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"><br></span></div><div id="yui_3_16_0_1_1450286497486_3700" class=""><span class="" id="yui_3_16_0_1_1450286497486_3498" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">Proportional completion completes incompletely ranked
ballots so as to preserve the version of the Condorcet matrix that uses matrix
elements of the form VA>B/(VA>B + VB>A), that is it preserves the
proportion of voters amongst those who have expressed an opinion.<span class="" id="yui_3_16_0_1_1450286497486_3500" style="mso-spacerun:yes"> </span>Proportional completion is achieved by
dividing up incomplete ballots proportionally.<span class="" id="yui_3_16_0_1_1450286497486_3502" style="mso-spacerun:yes">
</span>For example, if for all ballots that do have a ranking between A and B,
60% rank A>B and 40% rank B>A then each incomplete ballot that does not
provide a ranking between A and B is divided up so that 60% of it ranks A>B
and 40% ranks B>A.<span class="" id="yui_3_16_0_1_1450286497486_3504" style="mso-spacerun:yes"> </span></span></div>
<div id="yui_3_16_0_1_1450286497486_3506" class=""><span class="" id="yui_3_16_0_1_1450286497486_3508" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif"><br></span></div><div class=""><span class="" id="yui_3_16_0_1_1450286497486_3508" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">I define the Proportional Completion voting methods
criterion to be satisfied when a voting method produces the same result if
incomplete ballots are proportionally completed or not.</span></div>
</div>
<span class="" id="yui_3_16_0_1_1450286497486_3510" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-fareast-font-family:Calibri;mso-fareast-theme-font:minor-latin;color:#00000A;mso-ansi-language:EN-US;mso-fareast-language:EN-US;mso-bidi-language:AR-SA"><br class="" id="yui_3_16_0_1_1450286497486_3512" style="page-break-before:always;mso-break-type:section-break" clear="all">
</span>
<div id="yui_3_16_0_1_1450286497486_3514" class=""><span class="" id="yui_3_16_0_1_1450286497486_3516" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">Terminology:</span></div>
<div id="yui_3_16_0_1_1450286497486_3518" class=""><span class="" id="yui_3_16_0_1_1450286497486_3520" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">V_A is the total number of ballots that contribute to
electing A.<span class="" id="yui_3_16_0_1_1450286497486_3522" style="mso-spacerun:yes"> </span>Ballots are included in
which Candidate A is the highest ranked hopeful (not elected and not excluded)
candidate.</span></div>
<div id="yui_3_16_0_1_1450286497486_3524" class=""><span class="" id="yui_3_16_0_1_1450286497486_3526" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">S_A is the sum of the loads of all of the ballots that
contribute to electing A. </span></div>
<div id="yui_3_16_0_1_1450286497486_3528" class=""><span class="" id="yui_3_16_0_1_1450286497486_3530" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">V_act is the total number of active ballots.<span class="" id="yui_3_16_0_1_1450286497486_3532" style="mso-spacerun:yes"> </span>These are the ballots that rank at least one
hopeful candidate. </span></div>
<div id="yui_3_16_0_1_1450286497486_3534" class=""><span class="" id="yui_3_16_0_1_1450286497486_3536" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">V_inact is the total number of inactive ballots.<span class="" id="yui_3_16_0_1_1450286497486_3538" style="mso-spacerun:yes"> </span>These are the ballots that do not rank any
hopeful candidates.<span class="" id="yui_3_16_0_1_1450286497486_3540" style="mso-spacerun:yes"> </span>They only rank
elected and excluded candidates. </span></div>
<div id="yui_3_16_0_1_1450286497486_3542" class=""><span class="" id="yui_3_16_0_1_1450286497486_3544" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">S_inact is the sum of the loads of all of the inactive
ballots.</span></div>
<div id="yui_3_16_0_1_1450286497486_3546" class=""><span class="" id="yui_3_16_0_1_1450286497486_3548" style="font-size:12.0pt;line-height:107%;font-family:"Times New Roman",serif">V_tot is the total number of ballots.<span class="" id="yui_3_16_0_1_1450286497486_3550" style="mso-spacerun:yes"> </span>V_tot = V_act + V_inact.<span class="" id="yui_3_16_0_1_1450286497486_3552" style="mso-spacerun:yes"> </span>Unlike all of the other variables listed
above, it does not change throughout the procedure.</span></div>
<div id="yui_3_16_0_1_1450286497486_3554" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3556" style="font-size:12.0pt;font-family:"Times New Roman",serif"><br></span></div><div id="yui_3_16_0_1_1450286497486_3664" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3556" style="font-size:12.0pt;font-family:"Times New Roman",serif">Proof that the ranked Phragmen method (QPQ)
satisfies proportional completion.<span class="" id="yui_3_16_0_1_1450286497486_3558" style="mso-spacerun:yes"> </span></span></div>
<div id="yui_3_16_0_1_1450286497486_3560" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3562" style="font-size:12.0pt;font-family:"Times New Roman",serif">Without proportional completion, the load
to elect candidate A is </span></div>
<div id="yui_3_16_0_1_1450286497486_3564" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3566" style="font-size:12.0pt;font-family:"Times New Roman",serif">s_A= (S_A +1)/V_A</span></div>
<div id="yui_3_16_0_1_1450286497486_3568" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3570" style="font-size:12.0pt;font-family:"Times New Roman",serif">This is also the load that will now be on
each ballot that contributed to electing A.<span class="" id="yui_3_16_0_1_1450286497486_3572" style="mso-spacerun:yes"> <br></span></span></div><div id="yui_3_16_0_1_1450286497486_3754" class="" style="line-height:normal"><br><span class="" id="yui_3_16_0_1_1450286497486_3570" style="font-size:12.0pt;font-family:"Times New Roman",serif"><span class="" id="yui_3_16_0_1_1450286497486_3572" style="mso-spacerun:yes"></span></span></div><div id="yui_3_16_0_1_1450286497486_3770" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3570" style="font-size:12.0pt;font-family:"Times New Roman",serif"><span class="" id="yui_3_16_0_1_1450286497486_3572" style="mso-spacerun:yes"></span>With proportional completion, the same formula applies but V_A and S_A
are modified to:</span></div>
<div id="yui_3_16_0_1_1450286497486_3574" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3576" style="font-size:12.0pt;font-family:"Times New Roman",serif">V^pc_A<span class="" id="yui_3_16_0_1_1450286497486_3578" style="mso-spacerun:yes">
</span>= V_A + V_inact *( V_A/V_act)</span></div>
<div id="yui_3_16_0_1_1450286497486_3580" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3582" style="font-size:12.0pt;font-family:"Times New Roman",serif">and </span></div>
<div id="yui_3_16_0_1_1450286497486_3584" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3586" style="font-size:12.0pt;font-family:"Times New Roman",serif">S^pc_A = S_A + S_inact*(V_A/V_act).</span></div>
<div id="yui_3_16_0_1_1450286497486_3588" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3590" style="font-size:12.0pt;font-family:"Times New Roman",serif"><br></span></div><div id="yui_3_16_0_1_1450286497486_3771" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3590" style="font-size:12.0pt;font-family:"Times New Roman",serif">The proportionally completed load to elect
A is therefore </span></div>
<div id="yui_3_16_0_1_1450286497486_3592" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3594" style="font-size:12.0pt;font-family:"Times New Roman",serif">s^pc_a = (S_A + S_inact * (V_A/V_act)<span class="" id="yui_3_16_0_1_1450286497486_3596" style="mso-spacerun:yes"> </span>+1) / ( V_A + V_inact*(V_A/V_act))</span></div>
<div id="yui_3_16_0_1_1450286497486_3598" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3600" style="font-size:12.0pt;font-family:"Times New Roman",serif">= (V_act/V_tot) * s_A + S_inact/V_tot</span></div>
<div id="yui_3_16_0_1_1450286497486_3602" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3604" style="font-size:12.0pt;font-family:"Times New Roman",serif">The load with proportional completion is
just a scale and a shift of the load without completion.<span class="" id="yui_3_16_0_1_1450286497486_3606" style="mso-spacerun:yes"> </span>It is the same scale and shift for each
candidate.<span class="" id="yui_3_16_0_1_1450286497486_3608" style="mso-spacerun:yes"> </span>A scale and shift does not
change the relative order of the loads so the same candidates would be elected
in the same order. The same scale and shift applies to the load to determine if
a candidate should be excluded.<span class="" id="yui_3_16_0_1_1450286497486_3610" style="mso-spacerun:yes"> </span>So the ranked
Phragmen (QPQ) satisfies proportional completion.<span class="" id="yui_3_16_0_1_1450286497486_3612" style="mso-spacerun:yes"> - Ross Hyman </span></span></div>
<div id="yui_3_16_0_1_1450286497486_3614" class="" style="line-height:normal"><span class="" id="yui_3_16_0_1_1450286497486_3616" style="font-size:12.0pt;font-family:"Times New Roman",serif"> </span></div><div dir="ltr">
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