Circular Ties

[Blake Cretney]

This is a reply to the Blake Reply message.  I've substituted a more
meaningful subject.

Donald E Davison wrote:
> 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.

Perhaps I wasn't clear in my objection to it.  

If a majority of those expressing a preference rank A over B, then I
say that based on the ballots, a majority prefers A to B.

A Condorcet winner is defined as the case where a single candidate is
preferred by a majority to every other candidate in the ballots
provided.  If every candidate has a majority against it from some
candidate, then this is a circular tie.

So, if we have the following ballots

40 A B C
35 B C A
25 C A B

A>B 65-35
B>C 75-25
C>A 60-40

The point is that this is a circular tie.  It doesn't matter whether
you use a Condorcet completion method, IRV, one of your new methods,
or don't bother to find a winner at all.  There is still a circular
tie.

In the same way, if I have the votes

60 A B
40 B A

Then A is a 1st place majority winner.  It doesn't matter how you
tabulate the results, A is still the 1st place majority winner,
because it has a majority of the 1st place votes.

The definition of what constitutes a circular tie, or a Condorcet
winner is not dependant on the method.  You can't really reduce the
number of circular ties by reducing the value of lower choices, any
more than you can make spring come faster by shortening February.  All
you've done is changed the definition of circular ties to something
that doesn't happen as frequently.

>      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.

If you mean that a voter prefers his higher ranked candidates to the
lower ones, this is certainly true, but irrelevant.  If you mean that
the difference in preference between candidates is greater the higher
you are on the ballot, then this is an unwarranted assumption that
will likely have highly negative effects.

For example, consider that there are two major parties, called X and
Y.  Party X runs two candidates, while Y runs one.  You are an X
supporter, and also slightly prefer X1 to X2.  You would certainly
greatly prefer either X1 or X2 to Y, so you mark your ballot as
follows

1 X1 
2 X2 
3 Y

But under the interpretation you gave, we would assume that X1 is
greatly preferred to X2 and Y.  The result of this incorrect
assumption is likely to be a vote-splitting problem.  Have you tested
your methods against GITC?

But even if your assumption was always correct, I am greatly
surprised that you would declare that a greater strength of feeling
from the voter justifies greater weight from the method.  This
contradicts the majoritarian principles you seem to hold so firmly.

Consider that the voters level of preference for three candidates, on
a 0 to 100 scale is as follows:

    A    B   C
40  100  0    0
30  90   100  0
30  0    100  90
AVR 67   60   27

In this example, the average rating given is highest for A, even
though B is the 1st-place majority choice.  An advocate of average
ratings would claim that A should still win because the majority held
position in favour of B is not held very strongly.  The minority
position in favour of A, however, is.

It seems to me that once you decide to base your decisions on the
strength of beliefs, instead of just the number of people on each
side, this is the logical conclusion.  You should test CCV to see if
it passes the Majority Criterion, assuming you want it too.

I, however, don't think that belief strengths should be considered,
as they are in Average Ratings, or CCV, and indirectly in other
methods.  The people who come to the most bizarre conclusions usually
seem to hold them most strongly.  I am suspicious of getting people to
self-assess their probability of being right.

> 
>      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 

Where does the 1/3 figure come from?  Why can't we just find a winner
using a Condorcet Criterion method like Path Voting?
   http://www.fortunecity.com/meltingpot/harrow/124/path

Methods like Path Voting will always choose the Condorcet winner when
there is one, and also find a winner when there isn't.  Why should
Condorcet advocates be tempted to embrace a method which fails the
Condorcet Criterion, just because it too finds an answer when there is
a circular tie?

--snip--
> 
>      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

Are you saying that your opposition to Condorcet is based on the
presence of circular ties?  I'm trying to understand exactly what form
this "gain" will take.

> of the lower choices. I know of three way in which we can reduce the value
> of the lower choices.
> 
--snip--
>      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

This is based on the theory that Condorcet discourages people from
making lower choices.  This charge is often repeated by IRV advocates.
 I offered a refutation in the email that you recently forwarded to
this list.  I don't know what more I can say.  The belief seems based
on a simple mathematical error.

IRV advocates note that:
1.  In some situations your favourite candidate can be made to win by
only ranking him or her.

And misconstrue this to imply
2.  Ranking only your favourite candidate makes it more likely for
him or her to win.

This is an obvious mistake, but it gets repeated so frequently that I
have to think it is one of the main reasons that people choose IRV
over Condorcet.

It is just as silly as taking the true statement

1.  Ranking your favourite candidate lower on the ballot can cause
him or her to win in IRV.

And assuming that this implies
2.  In IRV, rank your favourite candidate as low as possible to make
it most likely that he or she will win.

>      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.

I still don't understand why circular ties pose any special problem
for Condorcet completion methods.  Maybe you see it as obvious, but
I'm afraid it isn't to me.  Please explain this point.  If this is
Condorcet's biggest flaw, I'd like to be aware of it.

---
Blake Cretney
See the EM Resource:  http://www.fortunecity.com/meltingpot/harrow/124