Gordon Hazen writes:
> >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.
Dear Gordon
So far so good, but what is the second decision situation (I am still
at a loss how to construct it and how to relate it to the if-then)
and how does the decision behavior follow that you indicated?
L1' promises a 25% chance at a lifetime of 30 years, 75% chance at
immediate death.
L2' promises a 20% chance at a lifetime of 45 years. 80% chance at
immediate death.
I think I would prefer L1 to L2 and L2' to L1', regardless of what
expected utility theory says. But how do I get this situation with
the diagnostic test?
> 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).
I still don't like this way of using the word "utility". It do not
require that the decision is made by computing a number (a "utility")
for each option and then to make the decision by simply comparing
these numbers. All I want is that the decision is based not only
on the mean, but also on the variance - and, of course, that it is
plausible and fits my intuition, something expected utility theory
in the sense we have been discussing here doesn't ;-). How this is
done is an open question.
> 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.
That's what I am saying.
> The only way to satisfy your requirement is to get away from "computing
> expected values" of u(x).
Well, the expected value could be useful in assessing options,
it is just not enough, that's all.
> But if we do so, then there is no basis for your
> claim that the values of u(x) must be "metric/quantitative".
Sure, once we refrain from doing additions, multiplications and stuff
like that and confine ourselves to comparisons, u(x) need not be
metric anymore - ordinal is enough. But we *are* doing additions and
multiplications in computing expected values in this strange theory
and then u must be metric.
> 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.
Yes, as I already pointed out in an earlier post, the problem may be
that I do not know enough about it. However, the impression I got of
this theory through this discussion really does not make me inclined
to spend much time on it. It looks like a strangely forced way to put
everything into one function, which through this approach looses all
intuitive semantics. Sorry.
> 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.
Thanks a lot. If I can find the time...
> 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).
Yes, this seems to be fairly close to what I have in mind, although
I doubt that an offset is acceptable, because it changes ratios.
> 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).
Which is exactly what I am saying.
> If u(x) is a von Neuman-Morganstern utility function (that
> is, E[u(X)] does correctly rank lotteries)
Just a side question: What does "correctly" mean here?
> 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.
Very strange. What are the semantics of these functions u and v and
their values? Is there any reasonable way for a subject to specify
these functions? I mean, they are input to the theory, right? I need
them to compute the "utility" of an option. If I do not know them,
I cannot use the theory to come up with a suggestion for a decision.
And in what sense can the theory then be prescriptive? Is it that
on some occasions it can be shown that a certain result follows
regardless of the function I choose, provided that the function
satisfies certain requirements made in the theory? (see also below)
Of course, it could still be a trial for a descriptive theory. Let's
see how a subject behaves and whether we can describe this behavior with
such a function. However, we already know that expected utility fails
in this respect.
> 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.
Technically, mathematically, yes. But what are the semantics of the
transformation? There seem to be fairly few restrictions and those
that are there seem to lead to counterintuitive results.
The main problem I see is this: Suppose you have a certain set
of outcomes, which are assessed with a measurable value function.
Then I construct two decision problems by grouping these outcomes in
two different ways into two options and then I am interested in the
decisions to be made "rationally" in these two cases. Obviously, the
decision has to take into account not only the expected value within
each option, but also the variance - we seem to agree on this.
In this theory, we try to do this by applying a transformation on
the measurable value function, then we compute expected values for
the transformed "utility" and decide on the option with the higher
expected value, right? If we were completely free in choosing a
transformation, we could reach any result we wanted. However,
if I understand this correctly, expected utility theory requires
that the the transformation I apply preserves the relative ordering
of the outcomes as it results from the measurable value function.
Is this correct and if it is, why this requirement? I mean, with
the transformation I introduce a piece of information (attitude
towards risk and variance) that is more or less unrelated to the
assessment of the outcomes, could actually change the ordering.
It is clear that we cannot leave the choice unrestricted, but this
restriction seems much too strong to me. However, maybe this is not
a requirement of the theory?
> And I doubt there is any other coherent way to
> formulate your requirement.
And I doubt that there is no other way to formulate my requirement ;-)
I have to admit, though, that I have no suggestion.
Regards,
Chris
