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<P align=left>James,<BR>Regarding the Alternative Vote (aka IRV) you wrote:<BR></P><FONT face=TimesNewRomanPSMT>
<P align=left>"It is used for elections to the lower houses of Australia and Ireland, for mayoral</P>
<P align=left>elections in England, and for local elections in about twelve American cities."</P></FONT>
<P align=left><BR>In Ireland it is used to elect the President. The Irish lower house uses multi-winner</P>
<P align=left>STV. UK mayoral elections mostly use the "Supplementary Vote". From Wikipedia:<BR></P>
<P align=left>"The <A class=mw-redirect title="Supplementary Vote" href="http://en.wikipedia.org/wiki/Supplementary_Vote"><FONT color=#0000ff>Supplementary Vote</FONT></A> system is used for all <A title="Elected mayors in the United Kingdom" href="http://en.wikipedia.org/wiki/Elected_mayors_in_the_United_Kingdom"><FONT color=#0000ff>mayoral elections in England and Wales</FONT></A>. <BR>Under this system voters express a first choice and (optionally) a second choice. If no <BR>candidate receives 50% of first choice votes, the top two candidates go to a second round. </P>
<P align=left>Voters whose first choice has been eliminated but whose second choice is one of the top two </P>
<P align=left>candidates have their second preference vote added to the first-round totals for the leading </P>
<P align=left>candidates."<BR></P>
<P align=left>Of course it is equivalent to IRV when there are three candidates, but is otherwise awful.</P>
<P align=left> </P>
<P align=left>Regarding MinMax in your paper you wrote:<BR><BR><FONT face=TimesNewRomanPSMT>"The winner is the candidate whose worst pairwise loss (if any) is least bad;..."</FONT></P><FONT face=TimesNewRomanPSMT>
<P align=left><BR>You don't define here how you measure "least bad". Later you give this:<BR></P><FONT face=TimesNewRomanPSMT>
<P align=left>"Minimax</P></FONT><I><FONT face=TimesNewRomanPS-ItalicMT>
<P align=left>To calculate the winner</P></I></FONT><FONT face=TimesNewRomanPSMT>
<P align=left>1. Form a pairwise matrix. Form the </FONT><I><FONT face=TimesNewRomanPS-ItalicMT>N </I></FONT><FONT face=TimesNewRomanPSMT>by 1 vector </FONT><B><I><FONT face=TimesNewRomanPS-BoldItalicMT>MAXBEAT</B></I></FONT><FONT face=TimesNewRomanPSMT>, where </FONT><I><FONT face=TimesNewRomanPS-ItalicMT>MAXBEAT</FONT><FONT face=TimesNewRomanPS-ItalicMT size=1>x </I></FONT><FONT face=TimesNewRomanPSMT>is the</P>
<P align=left>greatest number of votes against x in any pairwise contest, i.e. </FONT><I><FONT face=TimesNewRomanPS-ItalicMT>MAXBEAT</FONT><FONT face=TimesNewRomanPS-ItalicMT size=1>x</I></FONT><FONT face=TimesNewRomanPSMT>=max(</FONT><I><FONT face=TimesNewRomanPS-ItalicMT>PM</I></FONT><FONT face=TimesNewRomanPSMT size=1>:,</FONT><I><FONT face=TimesNewRomanPS-ItalicMT size=1>x</I></FONT><FONT face=TimesNewRomanPSMT>). The</P>
<P align=left>candidate with the smallest value in the </FONT><I><FONT face=TimesNewRomanPS-ItalicMT>MAXBEAT </I></FONT><FONT face=TimesNewRomanPSMT>vector is the winner."</FONT></P>
<P align=left><FONT face=TimesNewRomanPSMT></FONT> </P>
<P align=left><FONT face=TimesNewRomanPSMT>This make no reference to "pairwise losses", so isn't it "MinMax(Pairwise Opposition)" that<BR>*fails* the Condorcet criterion?</FONT></P>
<P align=left><FONT face=TimesNewRomanPSMT></FONT> </P>
<P align=left><FONT face=TimesNewRomanPSMT><A href="http://nodesiege.tripod.com/elections/#methmmpo">http://nodesiege.tripod.com/elections/#methmmpo</A></P>
<P align=left></FONT><BR> </P></FONT>
<P align=left>Chris Benham<BR></FONT><FONT face=TimesNewRomanPSMT size=1><BR></P></FONT></DIV></div><br>
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