[EM] Arrow's Theorem - The Return (again)

[Alex Small]

Eric Gorr said:
> I recently purchased a copy of Arrow's book 'Social Choice and
> Individual Values' Second Edition (ISBN: 0300013647).
>
> The last time this topic came up, it was argued that Arrow's Theorem
> only involved strict preferences, based on the document found at:

Eric-

I freely concede that I was wrong when I said that Arrow's Theorem assumes
strict preferences.  Arrow's Theorem allows for the possibility of equal
rankings.  I admit it.  And I admit that Reny makes an unnecessarily
restrictive assumption.  So let's remove Reny from further discussion.

However, the context of that argument was whether or not Approval Voting
is a ranked method in the sense of Arrow's Theorem.  And the answer is a
resounding NO!!!!!!!

Arrow assumes that the outcome is determined when voters report their
preferences, whatever they might be, or whatever the voters report them as
when they vote insincerely.  If there are 3 candidates I can vote A>B>C,
A=B>C, A>B=C, etc.

Approval Voting violates that assumption.  With 3 candidates Approval
Voting requires that I vote 2 of the candidates equal.  Now, if that's
what I was planning to do anyway (either because it's a sincere preference
or because of some strategic concern) then I'm in good shape.  But if I
was planning to sort them into 3 categories then Approval Voting says
"sorry, no can do."

Now, as I recall in our last exchange over this a great many photons were
shuffled back and forth over fiber optic cables as we argued this point. 
I don't know how to make it any clearer.

But I'm going to go even further and argue that CR is not a ranked method.
 I know, this will be controversial, but I'm in a reckless mood tonight. 
In the past 48 hours I've proven just how fragile 2D localization of
photons is relative to 3D.  I feel invincible.  So here goes.

Arrow's Theorem assumes that the social choice function is a mapping from
the set of voter preferences to the set of candidates.  So if I go into
the voting booth and fill out my ballot to indicate A>B>C (for instance)
that's all that matters.  Doesn't matter whether I say A=5, B=2, C=0, or
A=5, B=4, C=0.  Regardless of which grade ballot I submit, the machine
that reads the ballot should say "Ah, this ballot indicates A>B>C."  And
that piece of info should be sufficient to determine the outcome.

However, with CR the ballots A=5, B=2, C=0 vs. A=5, B=4, C=0 can produce
different election results.

Anyway, that's all I have to say about that.



Alex