Took me a while, but hope what I say is useful.
Jobst had good words, except he oversimplified.
Centuries ago Llull had an idea which Condorcet improved a bit -
compare each pair of candidates, and go with whoever wins in each
pair. Works fine when there is a CW for, once the CW is found, it
will win every following comparison.
BUT, in our newer studying, we know that there is sometimes a cycle,
and NO CW. Perhaps useful to take the N*N array from an election and
use its values as a test of Jobst's rules:
There may be some comparisons before the CW wins one. Then the
found CW will win all following comparisons.
BUT, if no CW, you soon find a cycle member and cycle members
win all following comparisons, just as the CW did above.
Summary:
We are into Condorcet with ranking and no approval cutoffs.
Testing the N*N array for CW is easy enough, once you decide
what to do with ties.
Deciding on rules for resolving cycles is a headache, but I
question involving anything for this other than the N*N array - such
as the complications Jobst and fsimmons offer.
Dave Ketchum
On Nov 17, 2009, at 8:53 PM, [email protected] wrote:
Here's a way to incorporate this idea for large groups:
Ballots are ordinal with approval cutoffs.
After the ballots are counted, list the candidates in order of
approval.
Use just enough randomly chosen ballots to determine the Lull winner
with 90%
confidence: let L(0) be the candidate with least approval. Then for
i = 0, 1,
2, ... move L(i) up the list until some candidate L(i+1) beats L(i)
majority
pairwise (in the random sample). If the majority is so close that
the required
confidence is not attained, then increase the sample size, etc.
Then with the entire ballots set, apply Jobst's Reverse Lull
method: Start with
candidate A at the top of the approval list. If a majority of the
ballots rank
A above the Lull winner (i.e. the presumed winner if A is not
elected) then
elect A. Otherwise, go down the list one candidate to candidate B.
Let L be the
top Lull winner with approval less than B. If a majority of ballots
rank B
above L, then elect B, else continue down the list in the same way.
In each case the comparison is of a candidate C with the L(i) with
the most
approval less than C's approval.
If the decisions are all made in the same direction as in the
sample, then the
Reverse Lull winner is the same as the Lull winner, but occasionally
(about ten
percent of the time) there will be a surprise.
If a voter knew that her ballot was going to be used in the forward
Lull sample,
she would be tempted to vote strategically. But in a large
election, most
voters would not be in the sample, so there would be little point in
them voting
strategically. If sincerity had any positive utility at all, it
would be enough
to result in sincere rankings (in a large enough election).
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