At 04:02 PM 8/21/2003 +0200, Christian Borgelt wrote: > > 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. > >It is definitely a very good property of expected utility theory to >do so. However, I doubt that there is no alternative, i.e., no other >possible theory that is also capable of avoiding such weird results.
The only proposal for such a theory that I am aware of is, I believe, the theory of expected regret (Bell 1982). >I also have to admit that I do not quite see how Allais' paradox allows >the behavior you describe. Let me take a guess at how you mean it: >The probability of the diagnostic test indicating high risk corresponds >to the 25% chance in the lottery version? Yes, but the remainder of your guess is not what I meant. Here is what I meant: L1 promises a sure lifetime of 30 years (Conservative treatment). L2 promises an 80% chance at a 45-year lifetime and a 20% chance at immediate death (Surgical treatment). >The diagnostic test would have to have some healing property No. The diagnostic test only indicates whether the patient is at high risk. It has no healing benefit. Regarding the second topic in my last message: > > 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. > >I don't like this description, because in my view it contains an >equivocation of the word "utility". See below. > > > 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. > >Indeed, this is a minimal, a necessary requirement. > > > But any increasing transformation v(x) = f(u(x)) will also satisfy > > v(x) > v(y) whenever x is preferred to y. > >Yes, v will satisfy *this* requirement, but maybe not some other >requirements, which u satisfies. > > > So v(x) could equally well be "the utility" of x. > >No, this does not follow! It follows only if you assume that all a >utility function is meant to express is a preference, nothing else. >However, such a view is not possible, as I see things. We are doing >computations with the values of the utility function. For example, >we are computing expected values. For this to be possible, the values >of the utility function cannot be seen as ordinal, but have to be >metric/quantitative. In addition, considering what utility is meant >to express, I would say that at least ratios of utilities matter >and therefore we cannot apply just any increasing transformation. >Actually, I see little else than the multiplication with a constant >factor that is admissible, if we want to preserve the semantics >intended by the subject who specified the values. I think you are confused. You want to require that the utility of X depend on more than just the mean of U = u(X). But if we are "computing expected values" to calculate the utility of X, that is, if the utility of X is E[u(X)], then of course the utility of X depends only on the mean of U. The only way to satisfy your requirement is to get away from "computing expected values" of u(x). But if we do so, then there is no basis for your claim that the values of u(x) must be "metric/quantitative". In a proper utility assessment, subjects rarely directly specify utility values. You really need to know more about utility assessment, the foundations of preference theory, and expected utility theory. Let me suggest you consult decision analyses texts that discuss these topics. For example Chapters 13,14 in Clemen (1996), Chapter 4 in Keeney and Raiffa (1976), or Chapter 4 in Raiffa (1968) on the foundations of expected utility and utility assessment; and Krantz et al (1971) on preference theory more generally. French (1986) is a great source if it is still in print. Citations below. > > Do you also want the utility of X to depend on more > > than just the mean of the random variable V = v(X)? > >This question is difficult to answer, because in my view there is >an equivocation of the word utility here: It is used to denote two >different things. First there is the utility of a specific outcome, >where there is no chance element involved. This is what I think the >utility function specifies. It is a subjective assessment made by >the person who has to make the decision. This person imagines >his-/herself in the situation and then says how much he/she values >it (relative to other situations). Then there is the "utility" of >the option, with the option leading to different possible outcomes, >so that it can be represented by a random variable over outcomes >(or the utilities of these outcomes). This, for me, has a different >character. It is possible, as you suggest, to define a "utility function" v(x) over sure payoffs x such that v expresses more than just preference and is different from the von Neuman-Morganstern utility function u(x). If we allow "strength of preference" as a primitive as well as preference, then under certain assumptions about strength of preference, we get a so-called "measurable value function" v(x) that reflects both preference and strength of preference, and is unique up to increasing linear transformation (i.e., w(x) = a*v(x) + b also measures strength of preference if a > 0). I think this might be close to what you have in mind when you speak about the "utility of a specific outcome, where there is no chance element involved". See Krantz et al (1971) or Dyer and Sarin (1982). A key point, however, is that such a measurable value function v(x) is not necessarily a von Neuman-Morganstern utility function, that is, one cannot necessarily use E[v(X)] to rank lotteries X (which may be exactly what you are saying). If u(x) is a von Neuman-Morganstern utility function (that is, E[u(X)] does correctly rank lotteries) then all one can say is that u and v rank sure payoffs identically. Therefore u and v are ordinally equivalent, that is, u(x) = g(v(x)) for some increasing function g. It is possible to require that the utility of a lottery X depend on more than just the mean of V = v(x), where v is a measurable value function. This is a coherent requirement because if w(x) = a*v(x) + b is an equivalent measurable value function, then the mean of W = w(X) is just a times the mean of V = v(X). But note then what happens if the utility of a lottery X is taken to be von Neuman-Morganstern utility E[u(X)]. We have E[u(X)] = E[g(v(X))] = E[g(V)]. If g is a nonlinear function, then indeed E[u(X)] *does* depend on more than just the mean of V. So expected utility satisfies your "more than just the mean" requirement, if that requirement is formulated using a measurable value function. And I doubt there is any other coherent way to formulate your requirement. Gordon References DE Bell, "Regret in decision making under uncertainty", Operations Research 30 (1982) 961-981. RT Clemen, Making Hard Decisions: An Introduction to Decision Analysis, Duxbury Press 1996 and later editions JS Dyer and RK Sarin, "Relative risk aversion", Management Science 28 (1982) 875-886. S French, Decision Theory: An Introduction to the Mathematics of Rationality, Ellis Horwood/ Halsted/ John Wiley 1986. DH Krantz, RD Luce, P Suppes, A Tversky, Foundations of Measurement, Academic Press, 1971. H Raiffa, Decision Analysis: Introductory Lectures on Choices under Uncertainty, Addison-Wesley, 1968. 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/
