[EM] Arrow's Theorem.
eric at ericgorr.net
Mon Jul 14 10:48:07 PDT 2003
At 10:20 AM -0400 7/14/03, Eric Gorr wrote:
>At 10:18 PM -0700 7/13/03, Alex Small wrote:
>>In my opinion, Arrow's theorem is more impressive when you have as few
>>assumptions as possible. When the list of incompatible assumptions is
>>large, somebody can say "Well, duh! If you pile on a whole bunch of
>>assumptions you're likely to make the task impossible."
I decided to write to Dr. Arrow concerning this and got the following
Dear Mr. Gorr,
Both statements are correct. The "monotonicity" condition together
with Independence of Irrelevant Alternatives, implies the Pareto
condition, which is the sufficient condition used in the "Vanderbilt"
version. Actually, the monotonicity condition is used in the first
statement of the theorem (first edition of my book, SOCIAL CHOICE AND
INDIVIDUAL VALUES, 1951), while I used the Pareto condition in the
second edition (1963). If one looks at the proof of the theorem in
the first edition, I showed that the monotonicity condition implied
the Pareto condition and then, in effect, derived the theorem from
the Pareto condition. The difference is, therefore, not very large.
So, it would appear that in all cases monotonicity is there even if
it is not mentioned explicitly.
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