Thinking and Deciding

WHY EXPECTED-UTILITY THEORY IS NORMATIVE 245
Similarly, if many people choose to have a surgical operation that they know has
a .0001 probability of causing death, in order to achieve some great medical benefit,
some of these people will die, but, if the utility of the medical benefit is high enough,
the extent to which all the people together achieve their goals will still be greater than
if none of them chooses the operation. When we add utilities across different people,
we are saying that the loss to some (a few, in this case) is more than compensated by
the gain to others (a great many, in this case).
The second point about adding utilities is that sometimes it does not make sense
to do so, because the utility of one outcome depends on the outcomes of other decisions.
To take a simple example, suppose I enter two lotteries, one with a prize of
a vacation in the Caribbean next month and the other with a prize of a vacation in
the Caribbean two months hence. The utility of the second vacation would be lower
if I won the first lottery as well. I would have had enough vacation for a while. To
avoid this problem of utilities not being constant as a function of the outcomes of
decisions, I can redescribe my decisions. Rather than thinking of this as two separate
decisions — whether to enter one lottery and whether to enter the other — I can think
of it as one decision. For this decision, there are four options — enter neither, enter
both, enter the first only, and enter the second only — and four possible outcomes
— win neither, win both, win the first only, and win the second only. The long-run
argument applies only to decisions that have been described in a way that makes the
utilities of outcomes independent of other outcomes. Because the long-run argument
applies across people, as we have seen, this can often be done.
The third point is that this whole approach is weak because it is true only if the
long run is very very long indeed, that is, infinitely long. Even when the argument is
applied to groups of people, the groups must be infinite in size. It may be possible to
save this sort of argument, but we do not need to do so.
The argument from principles
Another argument applies to each decision, rather than a series of decisions. It turns
out that expected-utility theory is implied by certain principles, or “axioms,” that are
closely related to the idea of rational decision making as whatever helps us achieve
our goals. These axioms create an internal consistency among the choices we would
make at a given time — something like the idea of coherence discussed in Chapter
5. This rather amazing fact was discovered by Ramsey (1931), 3 and the theory was
developed more formally by de Finetti (1937), Krantz, Luce, Suppes, and Tversky
(1971), Savage (1954), and von Neumann and Morgenstern (1947). The two main
principles are called weak ordering and the sure-thing principle.
I shall present these principles first in terms of “choice,” but the point of them is
that they are about utility or what Broome (1991) calls “betterness.” Think of “we
should choose X over Y” as meaning “X is better than Y.” In particular, X is better
3 Frank Plimpton Ramsey was a philosopher and mathematician who died at the age of 26, while he
was still a graduate student at Cambridge University, after making major contributions to scholarship.