[EM] Arrow's Theorem.

Alex Small asmall at physics.ucsb.edu
Tue Jul 15 10:58:20 PDT 2003

Eric Gorr said:
> It still seems to me that Approval does have preference ballots.
> They are simply binary preferences (either you like the option or you
> don't) from which an outcome (I assume this means winner) is uniquely
> determined.

True.  If all voters actually had binary preferences, then we could say
that Approval is a ranked method because it allows voters to fully express
their relative preferences.

However, because Approval does not allow for non-binary preferences, it
does not allow voters to express their complete preferences and is hence
not a ranked method in the strict sense.  Since Arrow assumes that voters
will, or at least can, have non-binary preferences, and he assumes that
the election method will make note of that, Approval does not satisfy
Arrow's assumptions.  Therefore, we cannot invoke Arrow to say that
Approval flunks IIA.  Instead, we have to examine voter behavior.

Suppose that the voters' preferences are:

30 A>B>C
21 B>A>C
24 C>B>A
25 C>A>B

The approval ballots are

30 AB
21 BA
24 CB
25 C

The vote totals are 51 A, 75 B, 49 C.  Now say that we remove C from the
race.  It's a two-way race, ad A beats B 55-45.  So voter behavior could
cause Approval to flunk IIA.

On the other hand, if the voters had in their minds some absolute ratings
of the candidates, and they only approved a candidate based on their
opinion of him alone, not based on his merits relative to others, then the
withdrawal of C would mean that all but the 24 C>B>A voters would sit it
out, and B would win.

Needless to say, the first scenario (voters changing their approval
ballots in response to candidate withdrawal) is far more likely to occur
in practice.

So, in conclusion:

Approval is not, in general, a preferential or ranked method.
Arrow's Theorem does not therefore cover Approval, so Arrow can't help us
draw any conclusions about Approval.
Under reasonable assumptions concerning voter behavior, Approval can flunk


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