Halpern in a previous e-mail described an interesting version of the
Ellsberg paradox (see below)
However appropriately viewed there appears to be no paradox. In the
following we provide a model that explains the decision choices in a
consistent way.
First we model the informaton about the possible outcomes using a
Dempster-Shafer belief structure.
Here we have 2 focal elements
f1 = {Red}
f2 = {Blue, Yellow}
the masses are: m(f1) = 30/90 = 1/3
m(f2) = 60/90 = 2/3
In REF #1 I introduced a technique for decision making when the
information
about the uncertainty is in the form of a D-S belief structure.
An important aspect of this approach is what I called "decision
attitude".
This is a value "alpha" taking a value in the unit interval. The value
1 being the most optimist and 0 being the most pessimistic with 1/2
being neutral.
In example A of Halpern using the method introduced in Ref#1 you get a
"valuation" for each choice (this a generalization of the idea of
expected
value)
V(Red) = (1/3) (1) + 2/3 (0) = 1/3
V(Yellow) = 1/3 0 + 2/3 Alpha = 2/3 alpha
We see if a person is pessimistic (alpha << 1/2) the choice is RED
In example B of Halpern using the method introduced in Ref 1 you get
valuation
V(RED or BLUE) = 1/3 (1) + 2/3 Alpha
V(Yellow or Blue) = 2/3
if a person is pessimistic (alpha << 1/2) the choice is Yellow or Blue
We see that this view provides a very consistent explanation of the
choices made by most people
MOST PEOPLE ARE SIMPLY USING A PESSIMISTIC DECISION ATTITUDE>
A key feature here is that the DM in valuating the alternatives is
assigning his "probabilities" based on the payoffs
REF#1:<fontfamily><param>Times</param><bigger><bigger>. Yager, R. R.,
"Decision making under Dempster-Shafer uncertainties," International
Journal of General Systems 20, 233-245, 1992.
</bigger></bigger></fontfamily>___________________________
HALPERN VERSION
The arguably more interesting version of Ellsberg's paradox has balls
of three different colors in the urn: 30 reds, and 60 that are some
combination of blue and yellow. A ball is drawn.
In situation A, you get to choose between betting on red and betting
on yellow (you get $1 if you guess right and 0 otherwise). In
situation B, you get to choose between between on red+blue or betting
on yellow+blue. (If you bet on red+blue, you get a dollar if the ball
drawn is either red or blue).
the second. That's inconsistent with having a subjective probability
on the balls (no matter what your attitude is to risk).
__________________________________________________________________
Ronald R. Yager
Machine Intelligence Institute
Iona College
New Rochelle, NY 10801
TEL: (212)249-2047 FAX:(212)249-1689 E-Mail: [EMAIL PROTECTED]
[EMAIL PROTECTED]
Home Page: http://www.panix.com/~yager/HP/rry.html
