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<DIV dir=ltr align=left><SPAN class=341430922-14122005><FONT face=Arial
color=#0000ff size=2>I derived those formulas a couple of years ago, and
verified them by hand through 5 alternatives. It was a major headache, and it
would not surprise me if I made a mistake. </FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=341430922-14122005><FONT face=Arial
color=#0000ff size=2></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=341430922-14122005><FONT face=Arial
color=#0000ff size=2>Rating all candidates equally is the same as "not voting"
only when you apply a counting-side rule. I don't mind ballots being discarded
because they don't affect the outcome, but in approval those ballots would
contribute to all the candidates' percentages, so IN GENERAL, on the collection
side it is better to not make such assumptions.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=341430922-14122005><FONT face=Arial
color=#0000ff size=2></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=341430922-14122005><FONT face=Arial
color=#0000ff size=2>(In my idealized model of stored ballots A=B=C>D=E=F is
how you'd record approval-style ballots).</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=341430922-14122005><FONT face=Arial
color=#0000ff size=2></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=341430922-14122005><FONT face=Arial
color=#0000ff size=2>I don't have a philosophical objection to using the
pairwise matrix on the counting side of the system at all. I just think when
comparing different Condorcet methods it would be helpful to have the original
voters' ballots, because when we get into hypotheticals like "if x B>A voters
had changed to B>C>A then..." we'd have the data on how likely that might
be.</FONT></SPAN></DIV><BR>
<BLOCKQUOTE
style="PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #0000ff 2px solid; MARGIN-RIGHT: 0px">
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<FONT face=Tahoma size=2><B>From:</B> election-methods-bounces@electorama.com
[mailto:election-methods-bounces@electorama.com] <B>On Behalf Of </B>rob
brown<BR><B>Sent:</B> Wednesday, December 14, 2005 3:50 PM<BR><B>To:</B>
honky98@aggienetwork.com<BR><B>Cc:</B> Election Methods Mailing
List<BR><B>Subject:</B> Re: [EM] number of possible ranked ballots given N
candidates<BR></FONT><BR></DIV>
<DIV></DIV>On 12/14/05, <B class=gmail_sendername>Rob LeGrand</B> <<A
onclick="return top.js.OpenExtLink(window,event,this)"
href="mailto:honky1998@yahoo.com" target=_blank>honky1998@yahoo.com</A>>
wrote:
<DIV><SPAN class=gmail_quote></SPAN>
<BLOCKQUOTE class=gmail_quote
style="PADDING-LEFT: 1ex; MARGIN: 0pt 0pt 0pt 0.8ex; BORDER-LEFT: rgb(204,204,204) 1px solid">Paul
Kislanko wrote:<BR>> The number of full ranked ballots is just the number
of permutations of<BR>> N alternatives = N!<BR>><BR>> If equal
ranknigs are allowed, it's N! + 2^N - 1<BR><BR>I assume you mean that there
are N! ways to arrange the candidates <BR>strictly, and then N - 1 spaces in
between that could hold either a > or<BR>a =, giving N! * 2^(N -
1). But I think that's an overestimate because<BR>it would count
both A=B>C and B=A>C. Did you mean something different?
</BLOCKQUOTE>
<DIV><BR>I did a quick check with 3 candidates, and his formula appeared to
give the correct answer (note that I eliminated redundant ones like your
example):<BR><BR>a=b=c<BR>a>b>c<BR>a>b=c<BR>a=b>c
<BR><BR>a>c>b<BR>a=c>b<BR><BR>b>a>c<BR>b>a=c<BR><BR>b>c>a<BR>b=c>a<BR><BR>c>a>b<BR>c>a=b<BR><BR>c>b>a<BR><BR>which
is 13, which agrees with <A
onclick="return top.js.OpenExtLink(window,event,this)"
href="http://www.google.com/search?q=3%21%2B2%5E3-1"
target=_blank>http://www.google.com/search?q=3%21%2B2%5E3-1</A><BR><BR></DIV></DIV>Although
i might not want to count a=b=c, which I would sorta consider "not
voting"...<BR><BR><SPAN class=gmail_quote></SPAN>
<BLOCKQUOTE class=gmail_quote
style="PADDING-LEFT: 1ex; MARGIN: 0pt 0pt 0pt 0.8ex; BORDER-LEFT: rgb(204,204,204) 1px solid">Paul
Kislanko wrote:<BR>And double yes to I'm glad someone's working on a way to
model the different methods from a universal ballot format. It should make
it easier to see the differences between Condorcet-based methods.
</BLOCKQUOTE>
<DIV><BR></DIV>Cool. While I may not be 100% aligned with your
philosophical objection to the matrix as an intermediate step, I am certainly
interested in exploring some methods which don't use it. (I'm not trying
to model the exisiting condorcet methods this way, but looking at other,
new[?] methods that don't use the matrix) <BR><BR>Unfortunately with 10
candidates, there are about 4 million possible unique ballots, so this
non-lossy compression scheme may be less helpful than I
hoped<BR><BR>rob<BR></BLOCKQUOTE></BODY></HTML>