[EM] Arrow's Theorem.

Eric Gorr 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."

Mystery Solved.

I decided to write to Dr. Arrow concerning this and got the following 
message back:

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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