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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;"> </span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">To everyone:</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">Below, Steve is considering the following section of the following June 21, 2016
</span><span lang="EN" style="font-family: "Times New Roman",serif; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">Wikipedia article:<span style="mso-spacerun: yes;">
</span>‘</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-font-kerning: 18.0pt; mso-bidi-font-weight: bold;">Monotonicity criterion’.<span style="mso-spacerun: yes;">
</span>He will refer to the author of that section as ‘Wiki’:</span></p>
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<b><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 18pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">Instant-runoff voting and the two-round system are not monotonic[</span></b><a title="Edit section: Instant-runoff voting and the two-round system are not monotonic" href="https://en.wikipedia.org/w/index.php?title=Monotonicity_criterion&action=edit§ion=1"><b><span lang="EN" style="color: blue; font-family: "Times New Roman",serif; font-size: 18pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">edit</span></b></a></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">Using an example that applies to
</span><a title="Instant-runoff voting" href="https://en.wikipedia.org/wiki/Instant-runoff_voting"><span lang="EN" style="color: blue; font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">instant-runoff
voting</span></a><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;"> (IRV) and to the
</span><a title="Two-round system" href="https://en.wikipedia.org/wiki/Two-round_system"><span lang="EN" style="color: blue; font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">two-round
system</span></a><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">, it is shown that these voting systems violate the mono-raise criterion. Suppose a
</span><a title="President" href="https://en.wikipedia.org/wiki/President"><span lang="EN" style="color: blue; font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">president</span></a><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">
were being elected among three candidates, a left, a right, and a center candidate, and 100 votes cast. The number of votes for an absolute majority is therefore 51.</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">Suppose the votes are cast as follows:</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><b><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Number of votes</span></b></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><b><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">1st preference</span></b></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><b><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">2nd preference</span></b></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">28</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Center</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">5</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">30</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Center</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">5</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">16</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Center</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">16</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Center</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">According to the 1st preferences, Left finishes first with 35 votes, Right gets 33 votes, and Center 32 votes,
thus all candidates lack an absolute majority of first preferences. In an actual runoff between the top two candidates, Left would win against Right with 30+5+16=51 votes. The same happens (in this example) under IRV, Center gets eliminated, and Left wins
against Right with 51 to 49 votes.</span></p>
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<b><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 13.5pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">[STEVE’S Additions:</span></b></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">S: Given these preferences, the Center candidate rather than the Right candidate should get eliminated because
he receives fewer 1<sup>st</sup> preference votes.<span style="mso-spacerun: yes;">
</span>Still, the principle of one-citizen-one-vote requires that the preference of each of these supporter be counted until a majority winner is discovered.<span style="mso-spacerun: yes;">
</span>Since 16 of them preferred the Left candidate next and 16 preferred the Right candidate next, the Left candidate receive a total of 51 votes and the Right candidate 49.<span style="mso-spacerun: yes;">
</span>Given both that only one candidate can win in this election and the principle of one-citizen-one-vote, no citizen would be able to sustain an objection to this result.</span></p>
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</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">1<sup>st</sup> IRV</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">COUNT</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Center</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">1</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">35</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">32</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">33</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">2</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">51</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";"> </span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">49</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">Wiki: But if at least two of the five voters who ranked Right first, and Left second, would raise Left, and vote
1st Left, 2nd Right; then Left would be defeated by these votes in favor of Center. Let's assume that two voters change their preferences in that way, which changes two rows of the table:</span></p>
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<p style="margin: 0in 0in 0pt; line-height: normal;"><b><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Number of votes</span></b></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><b><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">1st preference</span></b></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><b><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">2nd preference</span></b></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">3</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 2; mso-yfti-lastrow: yes;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">7</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
<font face="Times New Roman"></font></td>
</tr>
</tbody>
</table>
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 8pt; line-height: normal; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto;">
<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">Now Left gets 37 first preferences, Right only 31 first preferences, and Center still 32 first preferences, and
there is again no candidate with an absolute majority of first preferences. But now Right gets eliminated, and Center remains in round 2 of IRV (or the actual runoff in the Two-round system). And Center beats its opponent Left with a remarkable majority of
60 to 40 votes. </span></p>
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 8pt; line-height: normal; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto;">
<b><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 13.5pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">1-[STEVE’S IRV exploration:</span></b></p>
<font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font>
<table style="margin: auto auto auto 0.5in; mso-cellspacing: 1.5pt; mso-yfti-tbllook: 1184; mso-padding-alt: 1.5pt 1.5pt 1.5pt 1.5pt;" border="1" cellpadding="0">
<tbody>
<tr style="mso-yfti-irow: 0; mso-yfti-firstrow: yes;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;" colspan="3">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><b><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Wiki’s changed preferences (2<sup>nd</sup> Set of Ballots)</span></b></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><b><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Number of votes</span></b></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><b><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">1st preference</span></b></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><b><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">2nd preference</span></b></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">28</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Center</span></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 3;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">3</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 4;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">2</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 5;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">30</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Center</span></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 6;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">5</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 7;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">16</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Center</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 8; mso-yfti-lastrow: yes;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">16</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Center</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
<font face="Times New Roman"></font></td>
</tr>
</tbody>
</table>
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 8pt; line-height: normal; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto;">
<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;"> </span></p>
<font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font><font face="Times New Roman"></font>
<table style="margin: auto auto auto 0.5in; mso-cellspacing: 1.5pt; mso-yfti-tbllook: 1184; mso-padding-alt: 1.5pt 1.5pt 1.5pt 1.5pt;" border="1" cellpadding="0">
<tbody>
<tr style="mso-yfti-irow: 0; mso-yfti-firstrow: yes;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">2<sup>nd</sup> IRV</span></p>
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">COUNT</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Left</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Center</span></p>
<font face="Times New Roman"></font></td>
<td valign="top" style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">Right</span></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">1</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">37</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">32</span></p>
<font face="Times New Roman"></font></td>
<td valign="top" style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">31</span></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">2</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">40</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">60</span></p>
<font face="Times New Roman"></font></td>
<td valign="top" style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">0</span></p>
<font face="Times New Roman"></font></td>
</tr>
<tr style="mso-yfti-irow: 3; mso-yfti-lastrow: yes;">
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">3</span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";"> </span></p>
<font face="Times New Roman"></font></td>
<td style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";">60</span></p>
<font face="Times New Roman"></font></td>
<td valign="top" style="padding: 1.5pt; border: rgb(0, 0, 0); border-image: none; background-color: transparent;">
<font face="Times New Roman"></font>
<p style="margin: 0in 0in 0pt; line-height: normal;"><span style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman";"> </span></p>
<font face="Times New Roman"></font></td>
</tr>
</tbody>
</table>
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<p style="margin: 0in 0in 8pt; line-height: normal; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto;">
<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">S: These 28 make the Center candidate’s total of 60 (and the winner) while the 2 make
the total of the Left candidate 40.<span style="mso-spacerun: yes;"> </span>While it is true that this change in preferences changes the win for the Left candidate to a defeat, they still helped the Left candidate both by increasing the number of first preferences
he received and the average intensity of preference given to him.<span style="mso-spacerun: yes;">
</span>At the same time, he could not expect to win because he had received only a total of 40 to the total of 60 votes received by the Center candidate.</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">Consequently, currently I do not yet see this example as containing any anti-democratic element. As I see it, the
principle of ‘one-citizen-one vote’ only requires that each citizen’s vote (preferences) be counted equally as long as technically possible until the single-winner is discovered, i.e. the candidate who has received the highest intensity possible of majority
support.<span style="mso-spacerun: yes;"> </span>Accordingly, before the 2 preferences were changed by Wiki, the Left candidate won with 51 votes with an average intensity of 9.76 on a scale of 10.<span style="mso-spacerun: yes;">
</span>After the change, the Center candidate won with 60 votes with an average intensity of 9.53<span style="color: red;">.<span style="mso-spacerun: yes;">
</span></span></span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">Intensity of support calculations:</span></p>
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<u><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">1<sup>st</sup> Set of Ballots</span></u></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">35X10 +16X9=350+144=494</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">494 divided by 51=9.76</span></p>
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<u><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">2<sup>nd</sup> Set of Ballots</span></u></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">32X10 +28X9=320+252=572</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">572 divided by 60=9.53</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">In contrast, the intensity of support for a forced win by the Center candidate from the 1<sup>st</sup> Set of Ballots
would only be 9.36:</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">32 X 1<sup>st</sup> preferences and 58 X 2<sup>nd</sup> preferences for the Center candidate:</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">32X10+58X9=320+522=842</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">842 divided by 90= 9.36</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">S: Again, using IRV, the originally winning Left candidate was defeated after the author
changed the two voters’ preferences.<span style="mso-spacerun: yes;"> </span>This happened even though the Left candidate was now given two 1<sup>st</sup>, rather than two 2<sup>nd</sup>, preferences.<span style="mso-spacerun: yes;">
</span>Wiki sees this as a violation of the ‘</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN;">mono-raise criterion’<span style="mso-bidi-font-weight: bold;">:
‘giving higher preferences to a candidate should never harm him’.<span style="mso-spacerun: yes;">
</span>However, at least in some senses, these two higher preferences did help the Left candidate, i.e. they gave him two more 1st preferences. They also helped him by reducing the intensity of 2 preferences given to one of his competitors, i.e. by giving the
Right candidate two 2<sup>nd</sup> rather than two 1<sup>st </sup>preferences.<span style="mso-spacerun: yes;">
</span>By themselves, these changes would make it more likely that the Left candidate would win.<span style="mso-spacerun: yes;">
</span>It is only because more other citizens gave their 1<sup>st</sup> preference to the Center candidate over the Right candidate that the Right candidate correctly had to be eliminated in the second count.<span style="mso-spacerun: yes;">
</span>In turn, this appropriately required candidate Right’s 2<sup>nd</sup> preferences to be transferred:
<span style="mso-spacerun: yes;"> </span>28 to the Center candidate and 3 to the Center candidate.<span style="mso-spacerun: yes;">
</span>Given these two change made by these two citizens, the particular preferences given by the other voters required that the Left candidate not be elected.<span style="mso-spacerun: yes;">
</span>Consequently, it was not the isolated giving of two higher preferences to the Left candidate that ‘harmed’ him.<span style="mso-spacerun: yes;">
</span>It was how these preferences had to be combined with the preferences of all the other citizens that required the Left candidate to be defeated.<span style="mso-spacerun: yes;">
</span>This follows from the principle of one-citizen-one-vote.<span style="mso-spacerun: yes;">
</span>Again, g</span>iven both this principle and the fact that only one candidate can win in this election, no one would seem to be able to sustain an objection to this result.</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">S: Nevertheless, Wiki claims that IRV is not monotonic (i.e. violates the mono-raise
criterion).<span style="mso-spacerun: yes;"> </span>Does Wiki have Woodall’s definition of a
<span style="font-variant: small-caps;">mono-raise random</span> criterion (see below) in mind [Douglas R. Woodall Discrete Applied Mathematics 77 (1997) 81-98]?</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">In the light of the above discussion and the following definitions, I would very much
appreciate it if anyone could explain why they might still want to criticize IRV for being nonmonotonic.<span style="mso-spacerun: yes;">
</span>Is IRV fully democratic in the sense defined above?</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">Section 1.3 in Woodall’s article defines how a nonmotonic set of rules might ‘harm’
or ‘help’ candidate x:<span style="mso-spacerun: yes;"> </span>‘We shall say a candidate x is either
<i style="mso-bidi-font-style: normal;">helped</i> or <i style="mso-bidi-font-style: normal;">
harmed</i> by a change in the <i style="mso-bidi-font-style: normal;">profile</i> if the result, respectively, is to increase or decrease [the probability of electing x, i.e] P<sub>E</sub>(x). The following two properties are well known to hold for STV [which
reduces to IRV in a single-winner raced].</span></p>
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<ul style="list-style-type: disc; direction: ltr;">
<li style="color: rgb(0, 0, 0); font-size: 12pt; font-style: normal; font-weight: normal;">
<p style="color: rgb(0, 0, 0); line-height: normal; font-size: 11pt; font-style: normal; font-weight: normal; margin-top: 0in; margin-bottom: 0pt; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-add-space: auto; mso-list: l0 level1 lfo1;">
<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">Later-no-help</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">:<span style="mso-spacerun: yes;">
</span>Adding a later preference to a ballot should not help any candidate already listed.</span></p>
</li><li style="color: rgb(0, 0, 0); font-family: "Times New Roman",serif; font-size: 12pt; font-style: normal; font-weight: normal;">
<p style="color: rgb(0, 0, 0); line-height: normal; font-family: "Calibri",sans-serif; font-size: 11pt; font-style: normal; font-weight: normal; margin-top: 0in; margin-bottom: 8pt; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-add-space: auto; mso-list: l0 level1 lfo1;">
<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">Latter-no-harm</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">:<span style="mso-spacerun: yes;">
</span>Adding a later preference to a ballot should not harm any candidate already listed.’</span></p>
</li></ul>
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<p style="margin: 0in 0in 8pt; line-height: normal; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto;">
<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">Next, Woodall goes on to define nine different ‘versions of monotonicity.<span style="mso-spacerun: yes;">
</span>The basic theme is that a candidate x should not be harmed by a change in the profile that appears to give more support to candidate x.</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">Monotonicity:</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">1-(mono-raise)</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">
x is raised on some ballots without changing the orders of other candidates<span style="font-variant: small-caps;">;</span></span></p>
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<p style="margin: 0in 0in 8pt; line-height: normal; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto;">
<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">2-(mono-raise delete)
</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">x is raised on some ballots and all the candidates now below x on those ballots
are deleted from them;</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">3-(mono-raise random)
</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">x is raised on some ballots and the positions now below x are filled (or left
empty) in any way that results in a valid ballot;<span style="font-variant: small-caps;">
</span></span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">4-(mono-append)</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">
x is added to the end of some ballots that did not previously contain x<span style="font-variant: small-caps;">;</span></span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">5-(mono-sub-plump)</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">
some ballots without x are replace by ballots with x placed top and with no second choice<span style="font-variant: small-caps;">;</span></span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">6-(mono-sub-top)</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">
some ballots that do not have x placed top are replaced with ballots that do place x top (and are otherwise arbitrary)<span style="font-variant: small-caps;">;</span></span></p>
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<p style="margin: 0in 0in 8pt; line-height: normal; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto;">
<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">7-(mono-add-plump)
</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">further ballots are added that place x top and with no second choices<span style="font-variant: small-caps;">;</span></span></p>
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<p style="margin: 0in 0in 8pt; line-height: normal; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto;">
<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">8-(mono-sub-top)</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">
further ballots are added that place x top (and are otherwise arbitrary)<span style="font-variant: small-caps;">;</span></span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; font-variant: small-caps; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">9-(mono-remove-bottom)</span><span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">
some ballots are removed, all of which have x at bottom, below all other candidate<span style="font-variant: small-caps;">.’</span></span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">I look forward to any of your replies.</span></p>
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<span lang="EN" style="font-family: "Times New Roman",serif; font-size: 12pt; mso-fareast-font-family: "Times New Roman"; mso-ansi-language: EN; mso-bidi-font-weight: bold;">Steve</span></p>
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<p style="margin: 0in 0in 8pt;"><font face="Calibri"> </font></p>
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