Dear Lotfi, > 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?
I'll leave that question open to specialists on fuzzy set theory. If the theory is all that it's cracked up to be, I'll hope it can handle this question? (One obvious answer is to assume that all points in a random interval are equally probable, reducing it to a special case of the probability distribution interpretation.) > Basing the choice on expected > values trivializes what is intrinsically a complex decision problem. The fact that a proposed solution is simple can hardly be taken as criticism against it. Perhaps the problem is not as intrinsically complex as you think? > Furthermore, it revives the issue of questionable validity of > maximization of expected utility. The question was how standard decision theory would approach the stated problem. Since standard decision theory is based on maximising expected utility, the answer is bound to make use of it. This particular discussion is about whether the theory is able to address the type of problem you proposed. Whether the theory is valid is an entirely different discussion which we could of course resume, if you have fresh arguments to add to the recent debate on that. > The problem with existing bivalent-logic-based > probability theory is that it inherits from bivalent logic its > intolerance of imprecision and partial truth. Since probability distributions provide a quantitative description of imprecision and partial truth that in turn provides solutions to the problems you have posed, I am curious as to why you make this claim. sincerely, Konrad
