# [EM] Arrow's Theorem.

Alex Small asmall at physics.ucsb.edu
Tue Jul 15 11:33:14 PDT 2003

```Eric Gorr said:
> Would you agree that if we end up finding that equal ranking allowed
> that Approval is covered by Arrow's Theorem and that we could then  show
> that it fails?

OK, I'll dig up a source in a minute, but first let me explain my position
before I present evidence on my behalf:

If Arrow's Theorem considers methods that are mappings from the set of
voter preferences WHATEVER THOSE PREFERENCES MIGHT HAPPEN TO BE to the set
of candidates, then Approval is not covered by Arrow, because Approval
does not accurately make note of voter preferences when the preferences
are non-binary.  My understanding has always been that Arrow's Theorem
assumes that the voters' actual preferences are recorded on the ballots.
Approval does not allow voters to express their actual preferences if
their preferences are non-binary.  In that case, Arrow's conclusions do
not apply to Approval.  Approval has to be evaluated without the
assistance of Arrow's Theorem.

Also, what is the point of all of this?  To determine whether or not
Arrow's conclusion (IIA is impossible for preferential methods that
satisfy pareto and non-dictatorship when there are 3 or more candidates)
applies to Approval.  If Approval is not a preferential method AS DEFINED
IN ARROW'S THEOREM then the question of IIA needs to be evaluated by
examining voter behavior in examples, as I did in my previous post.

OK, here's a link to a paper that proves Arrow's Theorem.  The proof
assumes that voters may have strict preference orders (i.e. no equal
rankings).

http://faculty-web.at.northwestern.edu/economics/chung/mr/Reny.pdf

So, in conclusion:

Arrow's Theorem assumes that every voter is allowed to specify whatever
preference order he happens to have, and that the preference order CAN be
non-binary.

Approval only allows voters to sincerely express binary preferences, and
forces voters with non-binary preferences to insincerely rank some
candidates equal to one another.  But Approval gives voters considerable
flexibility when specifying an insincere "ranking."

Approval is therefore not in the class of methods examined by Arrow's
Theorem.  We therefore cannot use Arrow's Theorem to determine whether
Approval satisfies IIA.  We must look elsewhere when analyzing Approval.

Alex

```