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<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'>Evidently examples matter more than logic, so here’s
one last try in support of Jobst’s argument that pairwise counting
methods should have a pairwise collection method.<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'>Suppose I were a staunch “pro-life” believer, so
“anti-abortion” is my most important criterion. There are 5
candidates in the race, and A & E are both anti-abortion, but have opposite
views on gun control (A for, E against) and capital punishment (A against, E
for). B, C, and D are all pro-choice, and either pro gun control or
anti-capital punishment or both. When asked to rank all 5 I give
A>B>C>D>E.<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'>If you ask me to compare B, C or D to E I’d rank
E>any. If you ask me to compare B, C or D pairwise to each other, the
abortion issue isn’t a factor, and my sincere preference might be
D>either B or C because of fiscal policy and a virtual tie on the other “pro-life”
issues.<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'>To suggest that you can infer my sincere pairwise preference
between any two alternatives who are not my first choice among many is
unwarranted. <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'>On the other hand, if you give me the option to explicitly
express my sincere pairwise preferences, you can directly fill in the pairwise
matrix with my sincere preferences. You can even use my answers to fill out a “ranked
ballot” so long as you allow ties and truncation ( { X, Y, either, neither}
being my choices for each pair).<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 face=Arial><span style='font-size:10.0pt;
font-family:Arial'>To prove that construction of a pairwise matrix from ranked
ballots is always possible, I think you’d need to show inductively that
all orderings by any voter of N candidates will always be the same for those as
those obtained by asking each voter to order N+1 candidates (with respect to
the N original candidates). I believe a logical consequence of such a proof
would be contrary to Arrow’s theorem, and therefore is impossible. (Just substitute
“issue” for “individual” and “ranked ballot”
for “group” and the same logic applies).<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'> <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>In theory, there is no difference between theory and practice; In
practice, there is. - </span></font><font size=2><span style='font-size:10.0pt'>Chuck
Reid</span></font> <o:p></o:p></p>
<p class=MsoNormal><font size=2 face="Times New Roman"><span style='font-size:
10.0pt'>----------------------</span></font> <o:p></o:p></p>
<p class=MsoNormal><font size=2 face="Times New Roman"><span style='font-size:
10.0pt'>Paul Kislanko</span></font><o:p></o:p></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><o:p> </o:p></span></font></p>
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