<html><head><style type="text/css"><!-- DIV {margin:0px;} --></style></head><body><div style="font-family:arial, helvetica, sans-serif;font-size:12pt"><DIV>Marcus,</DIV>
<DIV> </DIV>
<DIV>I have some questions about your draft (dated 23 June 2009) Shulze method</DIV>
<DIV>paper, posted:<BR></DIV>
<DIV><A href="http://m-schulze.webhop.net/schulze1.pdf">http://m-schulze.webhop.net/schulze1.pdf</A></DIV>
<DIV> </DIV>
<DIV>On page 13 you define some of the ways of measuring defeat strengths,</DIV>
<DIV>two of which are "Votes For" and "Votes Against":</DIV>
<DIV> </DIV>
<DIV><snip></DIV>
<DIV> </DIV>
<DIV><FONT size=3>
<P>Example 5 (</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">votes for</I></FONT></FONT><FONT size=3>): When the strength of the pairwise defeat </FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">ef </I></FONT></FONT><FONT size=3>is measured by </FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">votes for</I></FONT></FONT><FONT size=3>, </FONT></P>
<P><FONT size=3>then the strength is measured primarily by the absolute number N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>] of votes for candidate </FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>.</FONT></P>
<P><FONT size=3>(N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>],N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>]) </FONT><FONT lang=ZH-TW size=3 face=Math3,Math3><FONT lang=ZH-TW size=3 face=Math3,Math3></FONT></FONT><FONT size=1>for </FONT><FONT size=3>(N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">g</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New
Roman"><FONT size=3 face="Times New Roman,Times New Roman">h</I></FONT></FONT><FONT size=3>],N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">h</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">g</I></FONT></FONT><FONT size=3>]) if and only if at least one of the following conditions is satisfied: </FONT></P>
<P><FONT size=3>1. N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>] > N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">g</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">h</I></FONT></FONT><FONT size=3>]. 2. N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>] = N[</FONT><I><FONT size=3 face="Times New
Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">g</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">h</I></FONT></FONT><FONT size=3>] and N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>] < N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">h</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">g</I></FONT></FONT><FONT size=3>]. </FONT></P>
<P><FONT size=3></FONT> </P>
<P><FONT size=3>Example 6 (</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">votes against</I></FONT></FONT><FONT size=3>): When the strength of the pairwise defeat </FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">ef </I></FONT></FONT><FONT size=3>is measured by </FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">votes against</I></FONT></FONT><FONT size=3>, <BR>then the strength is measured primarily by the absolute number N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>] of votes for candidate </FONT><I><FONT size=3
face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>. </FONT></P>
<P><FONT size=3>(N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>],N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>]) </FONT><FONT lang=ZH-TW size=3 face=Math3,Math3><FONT lang=ZH-TW size=3 face=Math3,Math3></FONT></FONT><FONT size=1>against </FONT><FONT size=3>(N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">g</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New
Roman"><FONT size=3 face="Times New Roman,Times New Roman">h</I></FONT></FONT><FONT size=3>],N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">h</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">g</I></FONT></FONT><FONT size=3>]) if and only if at least one of the following conditions is satisfied: <BR>1. N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>] < N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">h</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New
Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">g</I></FONT></FONT><FONT size=3>]. 2. N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">f</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>] = N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">h</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">g</I></FONT></FONT><FONT size=3>] and N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">e</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times
New Roman">f</I></FONT></FONT><FONT size=3>] > N[</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">g</I></FONT></FONT><FONT size=3>,</FONT><I><FONT size=3 face="Times New Roman,Times New Roman"><FONT size=3 face="Times New Roman,Times New Roman">h</I></FONT></FONT><FONT size=3>].</FONT></P>
<P><FONT size=3></FONT> </P>
<P><FONT size=3><snip></FONT></P>
<P><FONT size=3></FONT> </P>
<P><FONT size=3>I am a little bit confused as to the exact meaning of the phrase "the absolute number ..of </FONT></P>
<P><FONT size=3>votes for candidate E".</FONT></P>
<P> </P>
<P><FONT size=3>Does "the number of votes for E" mean 'the number of ballots on which E is ranked above<BR>at least one other candidate'?</FONT></P>
<P><FONT size=3></FONT> </P>
<P><FONT size=3>Or does it mean something that can be read purely from the pairwise matrix?<BR><BR>Does it mean 'the sum of all the entries in the pairwise matrix that represent pairwise votes for E'?<BR><BR>Do the two methods 'Schulze(Votes For)' and 'Shulze(Votes Against)' meet Independence</FONT></P>
<P>of Clones?<BR></P>
<P>I look forward to hearing your clarification.</P>
<P><BR><BR>Chris Benham</P>
<P> </P></DIV></div><br>
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