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<div class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman""></span><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman""></span><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman"">In December I proposed a proportional
multi-winner method <span style="mso-spacerun:yes"> </span>that satisfies a Condorcet property. The method is constructed from a single winner societal ranking method, such as</span><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman""><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman""> Schulze beat path or Tideman ranked pairs,</span> and a conventional proportional multi-winner method used to choose n winners from n+1 candidates, such as Woodhall QPQ or STV.<br></span>
</div><div class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman""> </span></div>
<div class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman"">However I have not proved that the method shares
the desirable properties of the societal ranking and proportional methods that
make it up.<span style="mso-spacerun:yes"> </span></span></div>
<div class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman""> </span></div>
<div class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman"">If the proportional method that makes it up satisfies Droop proportionality,
which is the case for QPQ and STV, is that also the case for the combined
method?</span></div>
<div class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman""> </span></div>
<div class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman"">If the societal ranking that makes it up is monotonic and clone independent,
which is the case for Schulze beat path and Tideman ranked pairs, then does the
combined method satisfy multi-winner generalizations of those single winner
properties?</span></div>
<div class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman""> </span></div><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman"">If it can be shown that the method satisfies Droop proportionality, monotonicity, and clone independence, then I think it would be preferable to conventional QPQ or STV. Otherwise not. </span><div class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal; color: rgb(0, 0, 0); font-size: 12pt; font-family: "Times New Roman","serif"; background-color: transparent; font-style: normal;"><br><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman""></span></div></div><div class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman"">The Method </span></div>
<div class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman"">Consider the set of all n+1 candidate elections for n seats.<span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span>For
the lowest ranked candidate in the societal ranking, if it is the loser of any
of the elections in which it is a candidate, then remove all elections in which
it is a candidate.<span style="mso-spacerun:yes"> </span>If instead, the
lowest ranked candidate wins every election in which it is a candidate, then
remove all election in which it is not a candidate.<span style="mso-spacerun:yes"> </span>Continue this process for the next lowest
ranked candidate in the societal ranking, etc, until all of the remaining
elections have the same set of n winners.<span style="mso-spacerun:yes">
</span>These are the n winners of the method.</span></div><div><div class="MsoNormal" style="margin-bottom: 0.0001pt; line-height: normal; color: rgb(0, 0, 0); font-size: 12pt; font-family: "Times New Roman","serif"; background-color: transparent; font-style: normal;"><span style="font-size:12.0pt;font-family:"Times New Roman","serif";mso-fareast-font-family:
"Times New Roman""><br></span></div>
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