Eric Gorr

2003-08-02 19:36:53 UTC

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.

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.

--

== Eric Gorr ========= http://www.ericgorr.net ========= ICQ:9293199 ===

"Therefore the considerations of the intelligent always include both

benefit and harm." - Sun Tzu

== Insults, like violence, are the last refuge of the incompetent... ===

----

Election-methods mailing list - see http://electorama.com/em for list info

== Eric Gorr ========= http://www.ericgorr.net ========= ICQ:9293199 ===

"Therefore the considerations of the intelligent always include both

benefit and harm." - Sun Tzu

== Insults, like violence, are the last refuge of the incompetent... ===

----

Election-methods mailing list - see http://electorama.com/em for list info