-----Original Message-----
From: [log in to unmask]
[mailto:[log in to unmask]] On Behalf Of Jonathan Barzilai
Sent: September-28-09 1:56 PM
To: [log in to unmask]; [log in to unmask];
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Subject: Re: [Jdm-society] Fw: Mathematical Foundations of Decision Theory
Bob,
My apologies for the delay in responding - I was out of town.
I listed quite a few shortcomings of utility theory. They apply to von
Neumann & Morgenstern utility axioms, Luce & Raiffa’s utility structure,
etc. In particular, I am not aware of a single scale construction in the
literature that enables the application of addition and multiplication to
scale values. As you know, years ago I asked for a proof of the
applicability of these operations. The response of the leaders of INFORMS’s
Decision Analysis Society has been less than scientific – you may have noted
that they are not circulating my messages. At any rate, the operations of
addition and multiplication are not applicable to utility scale values
produced in all versions of utility theory and the theory is flawed in
additional ways as well, including the contradiction in my paradox.
If I understand your examples, in both cases a physical (non-subjective)
probability is to be maximized. In that case the problem is characterized by
the maximization of a physical variable rather than the subjective variable
named utility. In other words, the utility problem was transformed into a
physical one. If my understanding is correct, you are saying that when
utility maximization does not depend on utility theory, it does not suffer
from the shortcomings of utility theory. Since this is, of course true, but
does not say much, could you clarify the question?
Jonathan
-----Original Message-----
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Sent: September-23-09 9:55 AM
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Subject: [Jdm-society] Fw: Mathematical Foundations of Decision Theory
I'd appreciate your considering the following special cases where expected
utility theory seems to bypass some of the issues you raised.
Specifically consider those cases where the expected utility of a decision
is equivalent to the probability of the decision achieving some goal.
(1) Sometimes the requirements for achieving the goal are known up front,
e.g., my boss might have given me a hundred goals and given be the goal of
winning a thousand dollars for her in a casino. If I am successful, I
get promoted. Otherwise I get fired.
(2) In other cases, the requirements are uncertain. For example, my boss
might send myself and a rival to the casino and intends to promote the
individual who wins more money for her. In this case, my goal is to win
more money than my rival. Since I don't know how much money my rival
will win at the casino, I don't know how much money I will need to win to
achieve my goal.
In either case, expected utility theory is equivalent to choosing that
decision which maximizes some probability. Furthermore since all the
required probabilities used in my example are based on casino
probabilities, these examples are not problematic for those who question
subjective probability. (Personally I'm a believer in subjective
probabilities but others might not be.) I interpret these examples as
indicating that there are some cases where expected utility theory is
immune from the foundational issues you raise.
If my interpretation is correct, then your arguments can be interpreted as
arguments about whether it is legitimate to extend expected utility from
the goal-oriented examples I mentioned to more general cases where goals
are not involved. (Since I'm a believer in expected utility theory, I do
believe this extension is OK but I'd like to see if we can establish some
common ground.) So I'd appreciate knowing whether or not you find any
foundational problems in the two goal-based examples which I have
provided.
Bob
----- Forwarded by Robert Bordley/US/GM/GMC on 09/23/2009 08:26 AM -----
"Jonathan Barzilai" <[log in to unmask]>
Sent by: [log in to unmask]
09/22/2009 02:39 PM
To
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Subject
[Jdm-society] Mathematical Foundations of Decision Theory
A pre-print is posted at http://scientificmetrics.com/publications.html
Jonathan Barzilai, ?Preference Function Modeling: The Mathematical
Foundations of Decision Theory,? pp. 1?37, 2009. To appear in Trends in
MCDA, José Figueira, Salvatore Greco, Matthias Ehrgott (Eds.)
Jonathan Barzilai, D.Sc.
Professor
Dept. of Industrial Engineering
Dalhousie University
P.O. Box 1000
Halifax, Nova Scotia
B3J 2X4
5269 Morris Street, Room 203
Tel: 902-494-3263
Fax: 902-420-7858
Email: [log in to unmask]
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