Dear Konrad:
Simplicity of Version 2 is deceptive because options A and B
are asymmetrical. (Version 2: If I choose A, I will get "approximately
a;" and if I choose B, I will get "approximately b" with probability
"approximately p," or "approximately c" with probability 1-
"approximately p," with a lying between b and c.)
If, as you do, "approximately X" is interpreted as a
probability distribution, then the choice is between a probability
distribution and a probability distribution of bimodal probability
distributions. i.e., a second-order probability distribution. If
"approximately X" is interpreted as an interval, then the choice is
between an interval and a probability distribution of random intervals.
How would you handle this interpretation? Basing the choice on expected
values trivializes what is intrinsically a complex decision problem.
Furthermore, it revives the issue of questionable validity of
maximization of expected utility.
Please note that I am not an opponent of probability theory.
Essentially, what I am critical of is basing probability theory on
bivalent logic. The problem with existing bivalent-logic-based
probability theory is that it inherits from bivalent logic its
intolerance of imprecision and partial truth. It is this intolerance
that severely restricts the ability of probability theory to deal with
problems in which imprecision and partiality of truth play important
roles--as they do in most real-world problems.
Sincerely,
Lotfi
--
Lotfi A. Zadeh
Professor in the Graduate School, Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720 -1776
Director, Berkeley Initiative in Soft Computing (BISC)
