Thinking and Deciding

EXPECTED-UTILITY THEORY 237
that probably did not exist 10,000 years ago and do not exist now in some people,
such as the idea that 1 hour spent watching TV is a direct cause of having 1 hour
less to spend on the homework that is due tomorrow (or an equivalent reduction in
sleep). We apply the same measure across different content. The concept of utility
as something we can treat in this way is an even newer idea, which may take a few
hundred years to become as generally accepted as the idea of time or longitude.
Can we be so precise about assigning numbers to amounts of good or goal
achievement? In the early days of navigation, estimation of longitude was imprecise,
but now it can be done within centimeters. The same was true when time was
measured with sundials. Can we ever be this precise about utility as we are now with
time and longitude? Probably not. Still, it is useful to suppose that there is a reality
to it, just as their is to your longitude as you read this.
We might try to escape this assumption by imagining that we could really only
measure utility within some interval. Within that interval, any two points would be
indistinguishable. This won’t work. Consider the following points:
A B C D E F
Suppose that A, B, and C, abbreivated A–C, were within the interval that makes them
indistinguishable. And the same for B–D, C–E, and D–F. But if this were true, we
could in fact distinguish A and B, because A would be outside of the interval B–D
and B would be in it. Likewise for distinguishing B–C because of the interval C–E.
Thus, there would in fact be no interval within which we could not distinguish one
point from another. It is difficult to think of how such intervals could exist (Broome,
1997).
As in the case of any judgment, of course, when two outcomes are very close
together, we will find it difficult to judge which is better, just as we find it difficult
to judge which of two different colored patches is brighter or more saturated (less
gray). If we make such judgment repeatedly (after forgetting our previous answers),
though, we will find that we tend to favor one more than the other. In sum, the idea
that utility is as precise as longitude — and as difficult to judge as longitude was to
the early navigators — is not as unreasonable as it might seem to be at first.
Expected-utility theory
We have seen that some of the tradeoffs we must make involve conflicts between
utility and probability. Simple examples are choices such as whether to live with a
disturbing health problem or risk surgery that will probably help but that could make
the condition worse; or whether to put money into a safe investment or into a risky
investment that will probably yield more money but could result in a big loss. A
decision of this sort is a gamble. Expected-utility theory deals with decisions that
can be analyzed as gambles. The problem with gambles is that we cannot know the
future, so we must base our decisions on probabilities.