Dear Gordon Let me start by clarifying the descriptive/prescriptive issue, because it is at the core of the discussion. If we only have the premise that a decision theory fails as a descriptive theory, it does not follow that this theory is not an acceptable prescriptive theory, you are perfectly right about that. However, if I add as a second premise that I see as perfectly rational what people are doing in a situation in which the theory fails, I have to reject it also as a prescriptive theory, because then it prescribes (at least in this specific situation) something I see as irrational.
More generally, the problem is this: If a decision theory prescribes some decision, which I see as inappropriate, I have two options: Either I can reject the theory as inappropriate or insufficient (this is what Prof. Zadeh is doing) or I can try to convince myself that I am wrong in believing that people are acting rationally in the situation in which it fails (this mailing list has seen several posts from people who said that they had done this). As I see things this discussion is about which of the two options is the better one. (Maybe we can apply expected utility theory to come up with an answer ;-) As with most such questions, in the end it may be a matter of belief how we decide. If I believe that people are often acting irrationally, I may be inclined to go for the second alternative. If I believe more strongly in the rationality of the human mind, I may prefer the second. In finding an answer, maybe it could help to consider the following: It is well known that many people misestimate the number of people in the famous birthday problem. (How many people are needed, so that the probability that two of them have the same birthday is higher than 50%?) Here it is easy to convince oneself that the original estimate given by many people is actually wrong. However, what is really convincing in this case is not that one does the computations (using probability theory), but that one can explain how the misestimate comes about: When estimating, people are usually inter-/extrapolating linearly, and since it is easy to assess the probability for 2 people and 366 people, a lot of people guess some number in the vicinity of 180. However, the actual function is highly non-linear. What I would like in the cases of Allais' and Ellsberg's paradox is something similar: Why do most people reject the answer of expected utility theory on first sight? What is it that they are doing wrong? Is there something like the linear interpolation in the birthday problem that does not fit the situation? Of course, this can also be turned around: If I reject the theory as inappropriate, I should try to find the point where it either neglects something or makes some implausible assumption or something like this. This is what I am trying to do. I still think that expected utility theory does not handle the variance of outcomes appropriately. Thank you very much for being so detailed in answering my question. However, I have some problems with this answer (some of which result from me not being specific enough in asking my question, I am sorry). I have to admit that I was suprised when you introduced the exponential utility function. Of course one can get a different relative position of the mean values if one scales the domain. That was not my point. How I meant my question is this: I assumed that the payoffs were not just numerical descriptions of the outcomes (like amounts of money), but already contained an assessment of the value of the outcomes for the subject that has to make the decision. That is, I assumed that they already represented the "utility" of the outcome. In this case, if I am not mistaken, the expected values of the normal distributions coincide with the expected utilities, right? Which means that expected utility theory would prescribe to go for option A, regardless of the values of c_1 and c_2. This is precisely the point at which I cannot accept expected utility theory, because I believe (as I said above, in the end it may be a matter of belief) that there can be rational grounds in such situations to choose option B, depending, of course, on the values of c_1 and c_2. And this is what I mean by saying that expected utility theory does not take into account the variances of the outcomes, because it seems to me that the rational grounds are closely connected to these variances. By the way: I did not claim that it would be sufficient for a decision theory to handle variances in this sense in order to be able to deal with the paradoxes. I just hold the opinion that it is necessary. Maybe something else is needed in addition. Finally, I would like to consider the following: It is clear that I can reconcile the theory and my rejection of a prescribed decision by saying that I assigned a wrong utility function. I have to admit that I entirely dislike this approach. With it the theory can always be saved, especially since the utility of an outcome is a subjective thing and thus cannot be measured objectively. We are free to fiddle around with the utility function until it gives the decision we want. However, this is not how it should be (and I am certain that you agree with me on this). The utility function is an input to the theory and cannot be changed. And once the utility function is fixed, we either have to accept the prescription the theory makes, or we have to reject the theory. With which we are back at the beginning. Best regards, Chris
