<HTML><FONT FACE=arial,helvetica><HTML><FONT SIZE=2 PTSIZE=10 FAMILY="SANSSERIF" FACE="Arial" LANG="0">Forrest Simmons wrote:<BR>
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>How good can we do with a simple rule of conversion of ranked ballots to<BR>
>approval ballots?<BR>
<BR>
>Given a set of preference ballots, let n1, n2, ... be the numbers of top<BR>
>rank votes for candidates c1, c2, ... , respectively. (For now assume that<BR>
>every ballot fully ranks the candidates.)<BR>
<BR>
>Use these numbers n1, n2, ... as weights in the weighted median method for<BR>
>determining the approval cutoff for each ballot.<BR>
<BR>
>That's it, except for a review of the "weighted median method for<BR>
>determining approval cutoffs."<BR>
<BR>
>Suppose the weights are w1, w2, ... . In the present case the w's are the<BR>
>n's . In Joe Weinstein's original proposal, the w's were the respective<BR>
>winning probabilities<BR>
<BR>
This is quite an interesting idea. From looking at a very limited number of close examples based on one voting scenario I would guess (with 3 candidate contests) that this would find the Condorcet winner (if there is one) about 99% of the time.<BR>
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I did manage to construct an example where this conversion method failed though.<BR>
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215 A>B>C approve A<BR>
169 A>C>B approve A<BR>
109 B>A>C approve BA<BR>
177 B>C>A approve BC<BR>
115 C>A>B approve C<BR>
215 C>B>A approve CB<BR>
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The Condorcet winner is B narrowly beating A and C.<BR>
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A v B 499 v 501<BR>
C v B 499 v 501<BR>
A v C 493 v 507<BR>
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The Approval winner is C ( A 493, B 501 and C 507).<BR>
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My guess, again, is that this conversion method will only fail in the closest of 3 candidates contest.<BR>
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David Gamble<BR>
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