Kristofer Munsterhjelm wrote:
> If we consider the votes as bullet votes, then we can expand to:
> 45: Able > Baker = Charlie
> 40: Baker > Able = Charlie
> 15: Charlie > Able = Baker
> which produces the matrix you gave above.
Able Baker Charlie
------- ------- -------
Able -- 45 45
Baker 40 -- 40
Charlie 15 15 --
OK, I was wrong when I said the cross-diagonal cells have to add up to
100. This way of accounting for tied rankings dictates otherwise.
Suppose, instead, we treat tied rankings as a half a vote for each
candidate:
22.5: Able > Bake > Charlie
22.5: Able > Charlie > Baker
20.0: Baker > Able > Charlie
20.0: Baker > Charlie > Able
7.5: Charlie > Able > Baker
7.5: Charlie > Baker > Able
Able Baker Charlie
------- ------- -------
Able -- 52.5 65.0
Baker 47.5 -- 62.5
Charlie 35.0 37.5 --
In another post in this thread, Raph Frank describes a third way of
representing tied rankings using proportions. Using the example above
instead of his example:
32.73: Able > Baker > Charlie
12.27: Able > Charlie > Baker
30.00: Baker > Able > Charlie
10.00: Baker > Charlie > Able
7.94: Charlie > Able > Baker
7.06: Charlie > Baker > Able
Able Baker Charlie
------- ------- -------
Able -- 52.94 75.00
Baker 47.06 -- 72.73
Charlie 25.00 27.27 --
In this example, Able is the Condorcet winner in all three matrices.
Several questions:
(1) Is this true in general, for all possible profiles? If there's a
Condorcet winner, is it always the same candidate no matter how you
treat tied rankings?
(2) Are there profiles containing cycles for which different
Condorcet-completion methods would give different winners depending on
how the tied rankings are represented?
(3) Going back to Dave Ketchum's original proposal that different voting
methods can be used in different subjurisdictions (e.g. states in the
case of NPV) and the matrices added together, could the method of
representing tied rankings ever affect the outcome in the jurisdiction
as a whole? I haven't tried to work this out, but intuitively it seems
to me that the answer is yes.
(4) I gather that Kristofer's procedure is the one most frequently used
in discussions of Condorcet. Is that true, and what is the history or
reasoning behind this?
Thanks,
Bob
Kristofer Munsterhjelm wrote:
Bob Richard wrote:
I'm obviously missing something really, really basic here. Can
someone explain to me what it is?
> Take it from the FPTP count and recount it
> into the N*N array by Condorcet rules ...
I still have no idea what this means. Here's an example:
Plurality result:
Able: 45
Baker: 40
Charlie: 15
Here's a (very naive) NxN matrix (fixed-width font required):
Able Baker Charlie
------- ------- -------
Able -- 45 45
Baker 40 -- 40
Charlie 15 15 --
But it's not a Condorcet count because we have, for example, no idea
how many of the Able voters prefer Baker to Charlie and how many
prefer Charlie to Baker. As a result, the pairs of cells above and
below the diagonal don't add up to 100. I still don't see how we can
"recount it into the NxN matrix by Condorcet rules".
Someone please show me the NxN matrix that Dave Ketchum would use to
combine these votes with the other votes that had been cast on ranked
ballots.
If we consider the votes as bullet votes, then we can expand to:
45: Able > Baker = Charlie
40: Baker > Able = Charlie
15: Charlie > Able = Baker
which produces the matrix you gave above.
That's the "consider bullet voters" idea. The other one is to count
the plurality vote locally, so you get:
100: Able > Baker > Charlie
which gives
A B C
A 0 100 100
B 0 0 100
C 0 0 0
and which could be used for any voting system. I think the first idea
is better, though.
--
Bob Richard
Marin Ranked Voting
P.O. Box 235
Kentfield, CA 94914-0235
415-256-9393
http://www.marinrankedvoting.org
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