Reply to Blake Cretney
Donald E Davison
donald at mich.com
Tue Feb 23 05:38:37 PST 1999
Dear Blake,
It appears that I have been less than clear in my presentation of
"Condorcet without Circular Ties. I will try to correct that. What follows
is a revision containing massive changes.
Sorry about that.
Regards,
Donald
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Greetings,
It would be nice if Condorcet did not have those circular ties. Maybe
something can be done to eliminaate them.
In my studies of circular ties I have observed that it only takes the
shifting of a small percentage of votes in order for Condorcet to go in and
out of the circular tie zone.
I have also observed that it is possible to leave the zone by reducing
the value of the lower choices. This observation suggests that we may be
able to always stay out of the zone if we decreased the value of the lower
choices for every election. We are justified in reducing the lower choices
because the voters themselves value the lower choices less than their most
preferred choice.
Consider the following list of reducing scales for lower choices:
1.0 1.0 1.0 Condorcet uses this scale - one third circular ties.
1.0 1.0 0.9
1.0 0.9 0.8
1.0 0.8 0.7
1.0 0.7 0.6
1.0 0.6 0.5
1.0 0.5 0.4
1.0 0.4 0.3
1.0 0.3 0.2
1.0 0.2 0.1
1.0 0.1 0.0
1.0 0.0 0.0 Plurality uses this scale - no circular ties.
Condorcet uses the top extreme scale. Plurality uses the bottom
extreme. Condorcet has as many as one third circular ties. Plurality has no
circular ties. The theory is that some scale between 1-1-1 and 1-0-0 is the
line between the circular tie zone and no circular ties. This line is at
0.5 for Blake's 2/22 example of a circular tie.
Blake's example: 40 ABC 35 BCA 25 CAB
Same as: 40A 40B 35B 35C 25C 25A
2nd choice X 0.5 40A 20B 35B 17.5C 25C 12.5A
Pairings 52.5A>35B 55B>25C 37.5<40A A is winner - no tie.
This one example shows that we can get out of the zone of circular
ties but this theory needs more testing. The use of a reducing scale is the
best way to test. I have worked a few examples, but a few examples do not
make a proof. It would be nice if someone, that had a program, would run
many examples.
About one year ago, Demorep ran a program on Condorcet in which he was
able to tell us how many circular ties we could expect. If Demorep could
run that program again, but this time using a Borda Count scale for the
lower choices, he may be able to tell us if the scale will take us out of
the circular tie zone.
If this theory proves to be true, the gain for the pairwise people
would be great - no circular ties. We will need a way to reduce the value
of the lower choices. I know of three way in which we can reduce the value
of the lower choices.
I know of three way in which we can reduce the value of the lower choices.
1) Use a reducing scale.
2) Make the lower choices netural.
3) Use a form of Cumulative Voting. (my favorite way)
If we used a reducing scale, like 10-5-4-3, for a four candidate
election, this would reduce the values of the lower choices and we may end
up outside the circular tie zone.
The second way to reduce the values is to make the lower choices
mathematically netural. This could also reduce the value of the lower
choices.
The first choices are netural because every candidate has the same
mathematical possibility of receiving 0 - 100 percent of the votes. The
lower choices are not netural because every candidate does not have the
mathematical possibility of receiving the same percent of the lower
choices.
For example, consider: 40 A 30 B 20 C 10 D
Candidate D has the mathematical posssibility of receiving 0 to 90
percent of the second choices, while candidate A has the mathematical
possibility of receiving only 0 to 60 percent of the second choices.
If we use factors to adjust the value of the second choices, we can
make the choices netural, and maybe we will also end up outside the zone of
circular ties.
The factor to use on the votes of candidate A will be 1.000 (60/60)
The factor to use on the votes of candidate B will be 0.857 (60/70)
The factor to use on the votes of candidate C will be 0.750 (60/80)
The factor to use on the votes of candidate D will be 0.667 (60/90)
(These factors only on this example of: 40 A 30 B 20 C 10 D )
This use of factors will give every candidate the same mathematical
possibility and also reduce the value of the lower choices, but this may
not be the best way to go, because this math will have to be repeated anew
for additional sets of choices when we have more than three candidates.
Condorcet with Cumulative Voting (CCV): I like this way the best for
reducing the value of the lower choices because it will be the voters that
decide how much value to give to each choice.
CCV would work as follows: The voter will be allowed a number of votes
equal to twice the number of candidates. The voter cast these votes the
same as he would in a Cumulative Voting election. The totals are then
worked using Condorcet. The following is an example:
100 A 40 B 80 B 30 A 50 C 20 A
75 A 30 C 50 B 10 C 30 C 15 B
---------- ---------- ---------
175 A 130 B 80 C
Pairs >>> A and B A and C B and C
175 130 175 80 130 80
20 15 30 10 40 30
--- --- --- --- --- ---
195 > 145 205 > 90 170 > 110
Note: There is a fourth way in which the count of the lower choices
may be reduced. This way will not require any rules and will happen anyway
and without design.
As more and more voters decide to only make one choice the Condorcet
method will be forced to function using lower and lower scales. When one
half of the voters are only making one choice, Condorcet will be operating
at about the one half value for the lower choices. If this happens to be
outside the zone of circular ties, then the voters will have solved the
circular tie problem without any effort on our part.
Again, I will say that this theory of reducing the value of the lower
choices to avoid the circular tie zone, needs testing to prove it right or
wrong. I will also say that those of you who favor Condorcet should do the
lion share of the testing, because if this theory proves to be correct, you
will have eliminated the biggest flaw of Condorcet.
Regards,
Donald
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| Q U O T A T I O N |
| "Democracy is a beautiful thing, |
| except that part about letting just any old yokel vote." |
| - Age 10 |
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