[EM] Arrow's Theorem.

Forest Simmons fsimmons at pcc.edu
Mon Jul 14 18:01:01 PDT 2003

On Mon, 14 Jul 2003, Rob Speer wrote:

> I will restate my question. I didn't know that people had such radically
> different ideas of Arrow's Theorem.
> I have seen Arrow's theorem defined with the following 4 criteria:
> 1. Preferential voting: Voters are allowed to express preference orders.
> 2. Non-dictatorship: There does not exist a single voter whose vote
> decides the outcome regardless of all other votes.
> 3. Pareto-optimality: if all voters prefer candidate A to candidate B,
> then candidate B is not the winner.
> 4. Independence from Irrelevant Alternatives: if a candidate is removed
> from the election, with the voters' relative preference orders among the
> remaining candidates being the same, and the removed candidate was not
> the winner, then the winner remains the same.

It is possible to interpret this fourth condition (IIAC) in two different
ways in the context of Approval.

You have interpreted it to mean that if a candidate is removed from the
election, the voters could not change their approval of the remaining
candidates, even if they wanted to in a way consistent with their
unchanging off-the-record utilities.

If they could, then in some cases they would:

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

If all voters vote sincerely, then the approval scores are 65, 75, and 60
for A, B, and C, respectively.

Remove the loser C and the 40 AB voters would want to demote B from
approved to disapproved.

Which would result in A as the new winner.

Note however that this result could not be inferred from the original
approval ballots, though it could be inferred from dyadic approval ballots
or the grade ballots used in "Five Slot Approval."

By the way, Approval made its appearance after Arrow's result became well
known.  I would surmise that Brams, et. al. were motivated by Arrow's
theorem, to find some way of getting around the hypotheses to avoid the
conclusion.  Bypassing the implicitly understood ballot style was a clever
way of getting around Arrow.

I agree with you that the weak point of Approval is that bad polls can
lead one to approve too many or too few candidates.  I like a method that
cannot be manipulated as much as Approval can be by fake poll results.
The Approval/Candidate Proxy hybrid that I have proposed is one solution.

Once the proxy approval ballots are added in to the non-proxy approval
ballots, the total set of ballots satisfies all of the nice properties of
approval including the IIAC (correctly interpreted), the consistency
criterion, and the FBC (favorite betrayal criterion).


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