Fw: A further elucidation of my arguments

[Paul Dumais]

David Catchpole wrote:
> 
> I'm afraid I can't except your argument along the lines of "Borda
> Probability" because, as Saari is apt to be paraphrased, it deliberately
> ignores a fact- in this case the fact of collective choice You're
> choosing a model of randomly selecting a ballot rather than assessing the
> wishes of an entire society, which is dangerously far from approaching
> the paradox I gave you.

I see an irrational fear of the word "probability". The reason I say
this is because my model does not at any time choose a ballot at random.
But perhaps I was not clear enough. The Borda Probability - BP(X) is a
measure which takes all ballots into account. The only reason its called
a probability is because the results are normalized. By normalization I
mean we divide a result by the total number of available votes. Using
probablty is valid here since it is equivalent to the following:
Candidates A and B are running. A gets 40 votes and B gets 60. We can
say that 60% of the people chose B. We could also say that if we choose
a vote at random, there is a 60% chance that that vote would be for B.
Notice I said "if". We are not implying that a vote such as this would
involve actually choosing a ballot at random to select the winner.
	Normalization is a good way to compare results of different votes where
different numbers of voteres are involved. If A and B run again next
month with 200 voters. If we said B got 65 votes this time we don't know
how he did against A unless we know how many votes were cast. By
normalizing we get a useful measure: 32.5% of voters chose B. This is
iquivalent to P(B) = 0.325 which is the probability that a randomly
chosen vote will be for B. The only reason I use probability theory is
that it opens the door to a very well established branch of mathematics
which can give an analytical breath of fresh air which this discussion
sorely needs. I think I have explained your paradox using a normalized
borda count or nanson method. If you have any more questions about
probabilty, you may save time by consulting a text book.

> Perhaps the thing that disturbsme most about Borda is that there is no
> theory to support it which actually reflect any of the reasons a
> particular voting system is better than another.

The above statement is only as good as the "reasons" you allude to. I'm
very open-minded however. I'm am eager to be proven wrong. To summarize,
I now prefer the Nanson method. It is easy to apply (just rank one or
more of the eligible candidates). It can be used for multiple seats.
There are no wasted votes. The result can be converted to a probability
(call it a ratio if you hate the word "probability") which can be used
to meaningfully compare elections with varying numbers of seats, voters
and candidates. I have not encountered an "unfair" result using this
method.

Criterions:

Consistancy: Can someone give me an example of how Nanson fails this?
GITC: I'm not sure how Nanson fails this iether.
Monotonicity: This criterion is not required to produce a fair result.
Since by changing some ballots such that X goes into lower positions you
put other candidates into higher positions. I'm not sure if Nanson can
fail without changing the relative postion of other candidates (which
would make the criterion inapplicable).
Reverse consistancy: I can't imagine an example where Nanson fails here.
SPC: This criterion doesn't seem to be necessary for a "fair" result.
LIIAC: Doesn't seem necessary for a "fair" result.

Thanks for the lively debate.


-- 
Paul Dumais