<div dir="ltr"><div><div>I said (in the second to the last paragraph below) that when there are only three candidates<br><br> <span><span></span>[X=Y>Bottom] =
[X=Y<Top],<br><br></span></div><span>What I meant was<br><br></span><span><span> </span>[X=Y>Bottom] =
[X=Y=Top],<br><br></span></div><span>This is because there will always be at least one top and one bottom candidate, which means there can never be two middle candidates in the case of only three total.<br></span><div><div><div><div class="gmail_extra"><br><div class="gmail_quote">On Thu, Dec 1, 2016 at 1:43 PM, Forest Simmons <span dir="ltr"><<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr">
<p class="MsoNormal">Kevin,</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">I’m not sure if you saw the email where I demonstrated the
equivalence of ICA and MDDAsc<span> </span>in the
case of implicit approval.<span> </span>So here is
the proof:</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">Both methods tart out by disqualifying “dominated” or “strongly
beaten” candidates unless that would eliminate all of them.<span> </span>Then they elect the most approved qualified
candidate.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">In MDDA “strongly beaten” means majority beaten.<span> </span>In MDDAsc <span> </span>it still means more than fifty percent of the
ballots opposed, but the “opposed” count takes symmetric completion of equal (unapproved)
rankings into account.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">In ICA (and ICT) “dominated” means the pairwise winning
margin is greater than the number of equal top rankings:<br><span> </span></p><p class="MsoNormal"><span><br></span></p><p class="MsoNormal"><span> </span>[X>Y]-[Y>X]-[X=Y=Top] > 0</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">If we add to this inequality the identity<span> </span>[X>Y] + [Y>X] +[X=Y] = 100% ,<span> </span>we get</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">2[X>Y] <span> </span>+
([X=Y]-[X=Y=Top]) <span> </span>> 100%</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">If we replace the difference in the parentheses with the
equivalent expression [X=Y<Top] <span style="font-size:11pt;line-height:115%;font-family:"calibri","sans-serif""><img height="20" width="92"></span><span>, and divide both sides
by two, we get</span></p><p class="MsoNormal"><span><br></span></p>
<p class="MsoNormal"><span>[X>Y]+</span><span style="font-size:11pt;line-height:115%;font-family:"calibri","sans-serif""><img height="27" width="6"></span><span>(1/2)[X=Y<</span><span style="font-size:11pt;line-height:115%;font-family:"calibri","sans-serif""><img height="20" width="11"></span><span>Top] = 50%.</span></p><p class="MsoNormal"><span><br></span></p>
<p class="MsoNormal"><span>This is precisely what we mean by
majority beaten with symmetric completion of equal rankings below Top.</span></p><p class="MsoNormal"><span><br></span></p>
<p class="MsoNormal"><span>So "strongly beaten" and "dominated"
mean<span> </span>the same thing in ICA/ICT and
MDDAsc as long as the symmetric completion affects all ranks below Top. <br></span></p><p class="MsoNormal"><span><br></span></p><p class="MsoNormal"><span>In
other words, ICT and MDDTsc are identical methods. <br></span></p><p class="MsoNormal"><span><br></span></p><p class="MsoNormal"><span>But the convention in MDDAsc
is that symmetric completion happens only among the unapproved candidates.<span> </span>If all ranked candidates are approved, then
symmetric completion takes place only at Bottom.</span></p><p class="MsoNormal"><span><br></span></p>
<p class="MsoNormal"><span>That’s why in my version of ICA candidate
X dominates candidate Y iff</span></p>
<p class="MsoNormal"><span>[X>Y] – [Y>X] > [X=Y>Bottom]</span></p><p class="MsoNormal"><span><br></span></p>
<p class="MsoNormal"><span>In the case of only three candidates we
always have<span> </span>[X=Y>Bottom] =
[X=Y<Top], so in that case my version of ICA is the same as yours. That is
why your simulations always give the same result for ICA and MDDAsc when approval
is taken as implicit approval.</span></p>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal"><span>In general the following definition of “dominates”
makes ICA precisely equivalent to MDDAsc no matter where the approval cutoff
is:</span></p>
<p class="MsoNormal"><span>Candidate X dominates candidate Y iff the
pairwise margin of defeat of Y by X is greater than the number of ballots on
which X and Y are both approved and ranked equal to each other.</span></p><p class="MsoNormal"><span><br></span></p>
<p class="MsoNormal"><span>My Best,</span></p><span class="gmail-HOEnZb"><font color="#888888">
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal"><span>Forest</span></p>
</font></span></div><div class="gmail-HOEnZb"><div class="gmail-h5"><div class="gmail_extra"><br></div></div></div></blockquote></div><br></div></div></div></div></div>