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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