[EM] Arrow's Theorem.

Rob Speer rspeer at MIT.EDU
Mon Jul 14 15:48:05 PDT 2003

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.

These criteria can be modified to become other forms of Arrow's Theorem
(for example, IIA can be described in terms of adding candidates instead
of removing them), but I'm fairly sure that the standard proof of
Arrow's Theorem works with just these criteria.

Now. I have _also_ seen Arrow's theorem defined with ONLY criteria 2-4 -
not mentioning voters being able to express preference orders.  In fact,
it has been defined that way frequently on this list. I believe this is
not true, because of Approval voting.

Here is why I believe that. I would like to know that I am wrong, but I
do not want people using their own convenient definitions.

Approval voting is non-dictatorial. If one voter approves of a
candidate, and everyone else disapproves, then that candidate will not
win, no matter who the voter is.

Approval voting is Pareto-optimal. The only way for every voter to
express that they prefer A to B is to approve of A and disapprove of B.
B has a 0% approval rating, and thus does not win.

Approval voting is independent from irrelevant alternatives. If a
candidate is removed from the ballot, and people's preferences remain
the same - they do not strategically change their votes - then the
winner will still win.
Rob Speer

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