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Warren,<br>
I have two main points in reply to your "DH3 pathology" anti-Condorcet
argument.<br>
<br>
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<caption>DH3 scenario with strategic votes by the A- and B-voters. </caption>
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<th>#voters </th>
<th>Their Vote </th>
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<td align="middle">37 </td>
<td align="middle">C>A,B>D </td>
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<td align="middle">32 </td>
<td align="middle">A>D>B,C </td>
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<td align="middle">31 </td>
<td align="middle">B>D>A,C </td>
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<p>Then the pairwise tallies are going to be: </p>
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<td>Definitely</td>
<td>A,B > D > C </td>
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<td>Probably</td>
<td>C > A,B </td>
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<p>In which case we (probably) have a Condorcet cycle scenario. (It
is actually two 3-cycles which share the common DC arc.) The weakest
defeats in these cycles are C>A,B which means, under both every
Condorcet rule I know of (since I think they all are equivalent in the
3-cycle case) and Borda, that one of {A,B} is going to be the winner. </p>
<p>I verified that A wins in the 50-50 mixture case under <a
href="RankedPairs.html">Tideman ranked pairs</a>, <a
href="SchulzeComplic.html">Schulze beatpaths</a>, and basic Condorcet
by using <a href="http://www.ericgorr.net/condorcet/">Eric Gorr's
Condorcet calculator</a> using this input </p>
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<pre>37:C>A>B>D
37:C>B>A>D
32:A>D>B>C
32:A>D>C>B
31:B>D>A>C
31:B>D>C>A
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<br>
The first is that those "defeat-dropper" style algorithms (like
Beatpath, Ranked Pairs, River,MinMax) that as you say are all
equivalent in the 3-cycle case<br>
are not my favourites. I prefer both DMC ('Definite Majority Choice',
which allows voters to enter approval cutoffs) and Schwartz,IRV (which
elects the<br>
member of Schwartz set highest ordered by IRV on the original ballots).
<br>
<br>
The latter has the property that a CW that is solidly supported by more
than a third of the voters (like C in your example) can't be
successfully buried,<br>
and the former is the same except that the CW must also be exclusively
approved by more than a third of the voters.<br>
So in the above example Schwartz,IRV elects C and if the C voters only
approve C so does DMC.<br>
<br>
My second point is that in your scenario the A and B supporters seem
mainly concerned to elect their favourites, so in that case why
wouldn't they simply<br>
be guided in their strategy by their favourite candidates? Seeing how
they stand in the polls, it would be in the interests of both A and B
to make a <br>
preference-swap deal at the expense of C. That way they each increase
their chances of being elected form below 33% to about 50% without
anyone<br>
having to flirt with the car-crash.<br>
<br>
Chris Benham<br>
<br>
<br>
Warren Smith wrote:
<blockquote cite="mid44F07D6C.mailECQ13OME7@euclid.math.temple.edu"
type="cite">
<pre wrap="">Sorry, for some reason, the hyperlink in my previous post was omitted.
Let me try again:
<a class="moz-txt-link-freetext" href="http://rangevoting.org/WinningVotes.html">http://rangevoting.org/WinningVotes.html</a>
This is a web page that discusses "winning votes" versus "margins" and
includes a careful look at the "DH3 pathology" which is a very serious
problem faced by both Borda and every Condorcet method. It is shown that
DH3 still clobbers Condorcet methods even if they allow equal rankings and
even if they use winnign-votes or margins.
The reason DH3 is "very serious" is it has the maximally bad effect (elects worst canddt)
and it has near maximal commonness (merely need 3 rivals and a dark horse running).
Perhaps you all will have some comments. I'm quite sure I have not included all
the wisdom about wv versus margin that EM has accumulated. But on the
other hand I also feel those at EM have underestimated or trivialized the power of
the DH3 pathology to cause massive destruction.
Warren D Smith
<a class="moz-txt-link-freetext" href="http://RangeVoting.org">http://RangeVoting.org</a>
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</pre>
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