<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>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal"><span>Forest</span></p>
</div><div class="gmail_extra"><br><div class="gmail_quote">On Thu, Nov 24, 2016 at 7:25 AM, Kevin Venzke <span dir="ltr"><<a href="mailto:stepjak@yahoo.fr" target="_blank">stepjak@yahoo.fr</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div><div style="color:#000;background-color:#fff;font-family:HelveticaNeue,Helvetica Neue,Helvetica,Arial,Lucida Grande,sans-serif;font-size:12px"><div id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31496"><span id="m_-5709259966056672732yui_3_16_0_1_1479996846138_32590"><font id="m_-5709259966056672732yui_3_16_0_1_1479996846138_32589" size="3">Hi Forest,</font></span></div><div id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31611"><span><font size="3"><br></font></span></div><div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31588"><span id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31703"><font id="m_-5709259966056672732yui_3_16_0_1_1479996846138_32588" size="3">I think references to MDDA's motivating criteria should be reduced because the new method MDDAsc shouldn't satisfy SFC or SDSC anymore.</font></span></div><div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31612"><span><font size="3"><br></font></span></div><div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31613"><span id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31984"><font id="m_-5709259966056672732yui_3_16_0_1_1479996846138_32587" size="3">I'm not sure it's natural to say that MDDA itself is a variant of ICA. I would say both are Condorcet//Approval variants that satisfy FBC.</font></span></div><div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31613"><span><font size="3"><br></font></span></div><div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31613"><span id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31983"><font id="m_-5709259966056672732yui_3_16_0_1_1479996846138_32585" size="3">I'm not sure if a different sort of reference to ICA would make enough sense... The apparent closeness between MDDAsc and ICA's results may be too obscure to be worth noting. For one thing, the similarity assumes that ICA uses an explicit approval cutoff, which isn't intended to be an option with ICA.</font></span></div><div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31613"><span><font size="3"><br></font></span></div><div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31613"><span id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31982"><font id="m_-5709259966056672732yui_3_16_0_1_1479996846138_32584" size="3">The draft doesn't mention the Plurality criterion. I'm not sure about it yet. My hunch is that to the extent that you aren't using symmetric completion below top, you're still vulnerable to whatever factors cause MDDA(implicit) to violate Plurality with 4+ candidates.</font></span></div><span class="HOEnZb"><font color="#888888"><div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31613"><span><font size="3"><br></font></span></div><div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31613"><span><font size="3">Kevin</font></span></div></font></span><div class="m_-5709259966056672732qtdSeparateBR" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31497"><br><div class="hm HOEnZb"><br></div></div><div class="m_-5709259966056672732yahoo_quoted" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31358" style="display:block"><div class="hm HOEnZb"> </div><div style="font-family:HelveticaNeue,Helvetica Neue,Helvetica,Arial,Lucida Grande,sans-serif;font-size:12px" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31357"><div class="hm HOEnZb"> </div><div style="font-family:HelveticaNeue,Helvetica Neue,Helvetica,Arial,Lucida Grande,sans-serif;font-size:16px" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31356"><div class="hm HOEnZb"> <div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31440"> <font id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31439" face="Arial" size="2"> <hr id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31614" size="1"> <b><span style="font-weight:bold">De :</span></b> Forest Simmons <<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>><br> <b><span style="font-weight:bold">À :</span></b> Michael Ossipoff <<a href="mailto:email9648742@gmail.com" target="_blank">email9648742@gmail.com</a>>; Chris Benham <<a href="mailto:cbenhamau@yahoo.com.au" target="_blank">cbenhamau@yahoo.com.au</a>>; Kevin Venzke <<a href="mailto:stepjak@yahoo.fr" target="_blank">stepjak@yahoo.fr</a>>; Forest Simmons <<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>> <br> <b><span style="font-weight:bold">Envoyé le :</span></b> Mercredi 23 novembre 2016 18h54<br> <b><span style="font-weight:bold">Objet :</span></b> Start of electowiki article on MDDAsc<br> </font> </div></div><div><div class="h5"> <div class="m_-5709259966056672732y_msg_container" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31355"><br><div id="m_-5709259966056672732yiv7599450795"><div dir="ltr" id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31354"><div id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31353"><div id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31352"><div id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31362"><div id="m_-5709259966056672732yui_3_16_0_1_1479996846138_31402">I've made a start on the electowiki article for MDDA(symmetric completion).<br><br></div>Please check my statements and examples for accuracy, and make suggestions for improved exposition, etc.<br><br></div>See the attachment.<br><br></div>Thanks,<br><br></div>Forest<br></div></div><br><br></div> </div></div></div> </div> </div></div></div></blockquote></div><br></div>