# [EM] Arrow and CR

Alex Small asmall at physics.ucsb.edu
Wed Oct 16 19:53:50 PDT 2002

```I was thinking about Arrow's theorem.  As I understand it, one statement
of the theorem is that you cannot construct a function of individual
preference orders that satisfies the conditions IIAC (Independence of
Irrelevant Alternative Candidates), Pareto (if all voters prefer A to B
then B should not win) and non-dictatorship.

Cardinal Ratings (of which Approval Voting is a special case, and some
would argue is a strategically equivalent case) does not, of course, use
preference orders as inputs.  Hence Arrow's Theorem does not strictly
apply to it, as I understand the matter.

Let's evaluate CR under the assumption that all voters vote sincerely,
i.e. if a voter prefers A to B then he or she assigns to B no more points
than she assigns to A.  We can also add the assumption that voters always
give their first choice the highest rating and their last choice the
lowest rating.

It certainly satisfies IIAC, if we implement IIAC by saying that voters
submit their ballots, and then we tally the results twice, once including
candidate C and once disregarding C.  The winner should be the same in
each case, unless C was the winner the first time.  (Of course, if voters
knew that C had been removed from the ballot they might adjust strategies.
That violates the standard ceteris paribis assumption of economics:  All
other things being the same.)

It satisfies Pareto, if we disregard the case of ties:  All voters prefer
A to B, and all voters give A and B equal ratings.

It obviously satisfies non-dictatorship.

Seems to me like we have an election method that satisfies Arrow's
criteria, even if the nature of the inputs doesn't satisfy Arrow's
Theorem.  Am I missing something?

Alex

----