I discussed briefly the "identification problem" that with certain
types of data, it is essentially impossible to distinguish without
further restrictions between expected utility and non-expected
utility decision makers. Further, essentially any decision rule
can be "rationalized" as optimal for an expected utility maximizer
with some utility function and beliefs. This result is related to
a result in game theory by Ledyard that there is no testable content
to the concept of Bayesian Nash equilibrium, i.e. essentially any
undominated strategy profile can be rationalized as a Bayesian Nash
equilibrium for some sets of preferences and beliefs. Kats Wakai asked
me for references on this after class: you can find the proof in my
Handbook of Econometrics chapter in section 3.5, The Identification
Problem, on page 3125. The chapter is online as a pdf file at
http://www.elsevier.com/hes/books/02/04/051/0204051.pdf
The chapter
also has a reference to Ledyard's article where his result is proven.
I also presented an example of the discrete choice decision rule and
the fact that with Type III extreme value errors, the conditional
choice probability resulting from this has the multinomial logit form.
The proof of this results from the fact that the extreme value family
of distributions is "max-stable", i.e. the maximum of a collection of
extreme value random variables also has the extreme value distribution
(this is analagous to the concept of the class of stable distributions
where the relevant operator is addition, e.g. the normal family is
stable since sums of normal random variables is normal). For the full
proof, see question 2 of problem set 5 for my Econ 551b course at
http://gemini.econ.yale.edu/jrust/econ551/exams/99/ps5/ps5.html
The
question guides you to the proof of the result via a more general
result called the Williams Daly Zachary Theorem. This theorem is the
discrete choice analog of Roy's Identity.
To give you something to do over the weekend, I assign this problem
as an exercise to be handed in at Tuesday's class. Also I ask you
to work through the details of example 2 of my lecture slide (page
6 in the slides at
http://gemini.econ.yale.edu/jrust/sdp/dplec.pdf
and include your derivation as part of this first assignment for the
course.
Otherwise, Have a good weekend!
Send questions/comments to: jrust@econ.yale.edu