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<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">A Condorcet divisor method proportional
representation procedure is presented that is a variant of Nicolaus Tideman’s Comparison
of Pairs of Outcomes by Single Transferable Vote (CPO-STV) and Shultz STV but
requires the determination of fewer candidate set comparisons than either.<span style="mso-spacerun:yes">  </span>The method will produce the same result as a party
list election that uses the same divisor method provided that each voter votes
their party’s list.<span style="mso-spacerun:yes">  The procedure is a Condorcet variant of the procedure presented in the February 2011 issue of Voting Matters.<br></span></span></p>

<p class="MsoNormal" style="margin-left:.15in;text-align:left;
text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">For an <i style="mso-bidi-font-style:normal">N</i>-seat
election, one primary election electing <i style="mso-bidi-font-style:normal">N</i>
candidates must be performed for each set of <i style="mso-bidi-font-style:
normal">N </i>+ 1 candidates.<span style="mso-spacerun:yes">  </span>For
example, for a two-seat election involving candidates A,B,C and D, primary
elections for the candidate sets ABC, ABD, ACD and BCD are held.</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">For each of these primary elections, the
winning set and its priority over loosing sets is determined by the following
procedure (the method is presented for the d’Hondt divisor method but is easily
generalized to other divisor methods.): </span></p>

<p class="MsoNormal" style="text-align: left; text-indent: 0in;" align="left"><span style="font-size: 12pt;" lang="EN-GB"><br></span></p><p class="MsoNormal" style="text-align: left; text-indent: 0in;" align="left"><span style="font-size: 12pt;" lang="EN-GB">Step 1. Every candidate in the <i style="mso-bidi-font-style:normal">N</i>+1 primary sub-election candidate set
is hopeful and every candidate not in that set is excluded.<span style="mso-spacerun:yes">  </span>The seat value of every ballot is set to zero. <br></span></p><p class="MsoNormal" style="text-align: left; text-indent: 0in;" align="left"><span style="font-size: 12pt;" lang="EN-GB"><br></span></p><p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">Step 2. The priority, <i style="mso-bidi-font-style:normal">P_C</i>,
for each hopeful candidate C that is the topmost hopeful candidate on at least
one ballot is determined from <i style="mso-bidi-font-style:normal">P_C</i>
= <i style="mso-bidi-font-style:normal">V_C</i>/(<i style="mso-bidi-font-style:
normal">S_C</i>+1) where <i style="mso-bidi-font-style:normal">V_C</i>
is the total number of ballots where C is the topmost hopeful candidate and <i style="mso-bidi-font-style:normal">S_C</i> is the sum of the seat
values of ballots where C is the topmost hopeful candidate.<span style="mso-spacerun:yes">  </span>The candidate with the highest priority is
elected. If the total number of elected candidates is N, the count is ended and
the <i style="mso-bidi-font-style:normal">N</i> elected candidates are declared
the winning candidate set of the primary with its priority over losing sets equal
to the priority of the <i style="mso-bidi-font-style:normal">N</i>th elected candidate.<span style="mso-spacerun:yes">  </span>Otherwise, if candidate C is elected, the
seat value for each ballot that contributed to electing C is increased to (<i style="mso-bidi-font-style:normal">S_C</i>+1)/<i style="mso-bidi-font-style:
normal">V_C</i>. Repeat Step 2 until N candidates are elected. </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">Each loser set from a primary contains the
candidate from the primary candidate set that is not in the winning set plus <i style="mso-bidi-font-style:normal">N</i>-1 additional candidates from the winning
set. For a two-seat election in which AB is the winning set of the primary
candidate set ABC, AC and BC are the loser sets.<span style="mso-spacerun:yes">  </span></span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">Only the priority of the winning set for
each primary is calculated.<span style="mso-spacerun:yes">  </span>The method
determines the priorities of fewer relations than Shultz STV but still elects
the Condorcet winner candidate set if there is one since the Condorcet winner candidate
set cannot be a losing set.<span style="mso-spacerun:yes">  </span></span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">Once every primary election has been held, winning
set > losing set relations are then elected from highest priority to lowest.
However, if electing a relation would violate transitivity then that relation is
excluded instead of elected.<span style="mso-spacerun:yes">  </span>In
practice, only loosing sets that are the winning set of at least one primary
election need be considered.<span style="mso-spacerun:yes">  </span>When every
relation has been elected or excluded, the highest ranked candidate set is
declared the elected candidate set.<span style="mso-spacerun:yes">  </span>An
example with a Condorcet cycle is the two-seat election presented in Election
1.</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">Election 1</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">7 A B C D</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">6 B C D A</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">5 C D A B</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">4 D A B C</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">Primary ABC</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">11 ABC</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">6 BCA</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">5 CAB</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">AB > AC and AB > BC. Priority: 8.5</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">Primary ABD</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">7 ABD</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">6 BDA</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">9 DAB</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">AD > AB and AD > BD.<span style="mso-spacerun:yes">  </span>Priority: 8</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">Primary ACD</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">7 ACD</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">11 CDA</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">4 DAC</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">CD > AC and CD > AD. Priority: 7.5</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">Primary BCD</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">13 B C D</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">5 C D B</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">4 D B C</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">BC > BD and BC > CD. Priority: 9</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">The winning sets are AB, AD, CD and
BC.<span style="mso-spacerun:yes">  </span>Since a candidate set must be a
winning set in at least one primary to win the election, only relations
involving winning sets need be considered.<span style="mso-spacerun:yes"> 
</span>The relevant candidate relations are</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">BC > CD. Priority: 9</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">AB > BC. Priority: 8.5</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">AD > AB.<span style="mso-spacerun:yes"> 
</span>Priority: 8</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">CD > AD. Priority: 7.5</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">AB > CD.<span style="mso-spacerun:yes"> 
</span>Priority: 6.5</span></p>

<p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB"> </span></p>

<p class="MsoNormal" style="text-align: left; text-indent: 0in;" align="left"><span style="font-size: 12pt;" lang="EN-GB">Transitivity can be preserved by electing
relations in priority order that preserve transitivity and excluding those that
do not.<span style="mso-spacerun:yes">  </span>When the three highest priority
relations are elected, they produce the transitive candidate set order AD >
AB > BC > CD.<span style="mso-spacerun:yes">  </span>The next highest
priority relation CD > AD is excluded since the higher priority relations have
determined that AD > CD.<span style="mso-spacerun:yes">  </span><span style="mso-spacerun:yes"></span>According to this procedure, candidates A and
D are elected.</span></p><p class="MsoNormal" style="text-align: left; text-indent: 0in;" align="left"><br><span style="font-size: 12pt;" lang="EN-GB"></span></p><p class="MsoNormal" style="text-align:left;text-indent:0in" align="left"><span style="font-size:12.0pt" lang="EN-GB">-Ross Hyman<br></span></p>

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