At 07:49 PM 8/20/2003 +0200, Christian Borgelt wrote: >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.
Christian, Yes, I agree. But I think you should take a look at my 8/5/2003 description of the Allais paradox before you conclude that the most common choices there are acceptable prescriptively. >...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? I think so. What people are saying in the Allais paradox is that they like L1 better than L2, but if faced now with 25% chance of having to choose between L1 and L2, they would prefer now that they would pick L2 if given the chance. Again, see my 8/5/2003 post for details. The problem is that people cannot see that they are doing this because they cannot multiply probabilities in their heads. Prescriptively, Allais paradox behavior is very hard to justify. It allows the following very odd behavior by a decision rule: 1) The decision rule prefers the policy "Choose surgery if the diagnostic test indicates high risk" to the policy "Choose conservative treatment if the diagnostic test indicates high risk". 2) The diagnostic test is conducted and indicates high risk. The decision rule now indicates that "Conservative treatment" is preferred to "Surgery". Expected utility is explicitly designed to avoid this weirdness. >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? So what you are saying is that if u(x) is the utility of sure payoff x (however defined), and X is a gamble over payoffs, then you want the utility of X (however defined) to depend not just on the mean of the random variable U = u(X), but also perhaps on the variance of U, or other features of the distribution of U. But there is a problem here: What is "the utility" u(x) of a sure payoff x? The function u(x) should satisfy u(x) > u(y) whenever x is preferred to y. But any increasing transformation v(x) = f(u(x)) will also satisfy v(x) > v(y) whenever x is preferred to y. So v(x) could equally well be "the utility" of x. Do you also want the utility of X to depend on more than just the mean of the random variable V = v(X)? Do you want this for every increasing transformation v of u? It is far from clear this is even possible. So I think your requirement is basically incoherent unless you further clarify what it is you want. Here is a further difficulty: In expected utility theory, the utility of a gamble X is E[u(X)], equal to E[U], where U is the random variable u(X). So the utility of X depends only on the mean of U, not its variance. But suppose we postulate that "the utility" v(x) of a sure payoff x is given by v(x) = exp(u(x)). There is nothing wrong with this, as v(x) is simply an increasing transformation of u(x). Then E[u(X)], which is still the utility of X, is given by E[ln(v(X))] = E[ln(V)], where V is the random variable v(X). And voila! the utility of X, being equal to E[ln(V)], will depend on more than just the mean of V, just as you wished. (For example, if U happens to be normally distributed, then V will have a lognormal distribution, and E[ln(V)] will depend on both the mean and variance of V.) So all you need to do to make expected utility acceptable according to your criterion is say that "the utility" of a sure payoff x is exp(u(x)), where u(x) is what everyone else calls the utility of x. Of course this is silly, but the reason it is silly is that your requirement is incoherent. Gordon Gordon Hazen Department of Industrial Engineering and Management Sciences McCormick School of Engineering and Applied Science 2145 Sheridan Road Northwestern University Evanston IL 60208-3119 Fax 847-491-8005 Phone 847-491-5673 Web: www.iems.nwu.edu/~hazen/
