[EM] Arrow's Theorem - The Return (again)

[Eric Gorr]

I recently purchased a copy of Arrow's book 'Social Choice and 
Individual Values' Second Edition (ISBN: 0300013647).

The last time this topic came up, it was argued that Arrow's Theorem 
only involved strict preferences, based on the document found at:

   http://faculty-web.at.northwestern.edu/economics/chung/mr/Reny.pdf

However, this does not appear to be the case.

On page 13, Arrow states as Axiom I

  For all x and y, either x R y or y R x

For Arrow's purposes, R is defined to mean in the case of x R y that a 
person either preferrs x over y or is indifferent in the comparison of 
x to y. To quote Arrow again,

   "Note that Axiom I is presumed to hold when x = y, as well as when x 
is
    distinct from y, for we ordinarily say that x is indifferent to 
itself
    for any x, and this implies x R x." (page 13)

After a few more formal statements and descriptions, based, in part, on 
Axiom I, he goes on to say that:

   "However, it may be as well to give sketches of the proofs, both to 
show
    that Axiom I and II really imply all that we wish to imply about the
    nature of orderings of alternatives and to illustrate the type of
    reasoning to be used subsequently." (page 14)

On page 36, he goes on to develop the Pareto principle based, in part, 
on Axiom I.

   "The Pareto principle was originally given in the text (p.36) as a 
form of
    the compensation principle." (page 96)

So, (to ask a question) why does Reny use only strict preferences?
Well, Arrow does the same thing near the end of his book, starting on 
page 96, he reasoning appears to be:

   "Since the Pareto principle is universally accepted, the new set of
    conditions will be easier to compare with other formulations of the
    problem of social choice." (page 96)

   "We give it [Pareto principle] here in a slightly weaker form 
(involving
    only strict preferences)." (page 96)

I would venture to guess that Reny used the same basic philosophy in 
his paper when writing about both Arrow's Theorem and 
Gibbard-Satterthwaite theorem.


Now, I fully admit that this may be a case of knowing just enough to 
get me into trouble and would appreciate it if someone else could take 
a look at the book and verify or clarify what I have said based on the 
original work.