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Rob Brown wrote:<br><br>
<blockquote type=cite class=cite cite>Well, I have nothing against using
beatpaths if the scores really do represent the way the results are
tabulated </blockquote><br>
They do represent the way the results are tabulated, but I don't think
that, in general, the "scores" that beatpaths produce will be
satisfying. For example, if you take<br><br>
49% Bush<br>
12% Gore>Bush<br>
12% Gore>Nader<br>
27% Nader>Gore<br><br>
A beatpath-score approach would give both Bush AND Nader a score of 27,
since this is the strength of the only defeat Gore suffers. While
this seems right to me for Nader, I think giving Bush a score of 49 seems
more right.<br><br>
<blockquote type=cite class=cite cite>(except that I must admit, I've yet
to quite understand how to work that method into my code .. I will look
into it further unless I first hit upon something that is really really
easy to code and works really well :) ). </blockquote><br>
I would again direct you to Rob LeGrand's website. The description
of "Schulze" at
<a href="http://cec.wustl.edu/~rhl1/rbvote/desc.html" eudora="autourl"><font color="#0000FF"><u>http://cec.wustl.edu/~rhl1/rbvote/desc.html</a></u></font>
is pretty easy to follow for someone who understands Condorcet (I think). You can also e-mail either Rob or Eric Gorr, as both have implemented beatpath.<br><br>
<blockquote type=cite class=cite cite> I have heard your suggestion of instead using the pairwise score against the winner, but don't feel it meets my criteria of being consistant with the logic used for determining ranking order. </blockquote><br>
I respectfully disagree. Think about it this way. When there is a Condorcet winner (a "beats all" candidate), then every candidate lost by their margin against the winner. Where they finished relative to everyone else is pretty much irrelevant, since they couldn't beat the winner. Condorcet is clone-independent after all.<br><br>
(When there is a circular tie, there simply isn't any 1-D set of scores that can accurately represent what is going on. So the performance of a scalar measure in this situation will never really be "perfect".)<br><br>
<blockquote type=cite class=cite cite>I prefer not have something that is "use this method, unless a certain case happens that causes a noticable conflict and then fall back on a completely different system".</blockquote><br>
Personally, I would be happy with a scoring system that always used every candidate's pairwise count against the winner, and allowed the winner to have a lower score than someone else in the case of circular ties. After all, when there's a circular tie there's really no set of 1-D scores that can represent that cycle. I suspect that you (and more to the point, your client) would not be satisfied with this. So I proposed the beatpath approach as a fallback in this case.<br><br>
Here's another potential technique that avoids beatpaths.<br><br>
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1) Calculate the winner and the Smith set using ranked pairs.<br><br>
2) For all alternatives outside the Smith set, their score is the number of votes they received pairwise against the winner.<br><br>
3a) If the Smith set contains only one alternative (i.e. the Condorcet winner), then the winner's score is the winning alternative's pairwise vote total against the losing alternative with the highest score (as defined in step 2).<br><br>
3b) If the Smith set contains more than one alternative (i.e there is a cyclic tie) then find the alternative within the Smith set that the winner has its strongest victory against (in terms of winning votes). Score every alternative in the Smith set based on their pairwise vote totals against this alternative. Assign this alternative a score equal to its pairwise votes against the winner.<br><br>
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3b basically amounts to breaking the cyclic tie in the fashion that guarantees the winner its best possible score. A simple approach, and one that I think gives very intuitive results.<br><br>
-Adam<br><br>
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