Thinking and Deciding

242 NORMATIVE THEORY OF CHOICE UNDER UNCERTAINTY
changing the unit) would just multiply the difference between EUA and EUB by the
same constant, but its direction would not change. If A were higher (better) than B,
it would still be better.
Pascal advocated this sort of analysis as what we would call a normative theory
of decision making in general. Implicit in his approach is the decomposition of decisions
into states, options, and outcomes (or consequences). To each state we can
assign a number expressing our personal probability judgment, and to each outcome
we can assign a number representing its utility. Once we have made these assignments,
we can calculate the expected utility of each option (or compare the expected
utilities of two options, as in 16.3) by a method completely analogous to the calculation
of the expected value of gambles.
A numerical utility estimate, as used in this kind of analysis, is not something that
exists in the head, to be read off as if from a thermometer. It is, rather, a judgment of
the desirability of an outcome, made (ideally) as if the judge knew all relevant facts
about the outcome. Utility judgments are useful for making important decisions.
We must remember, however, that expected-utility theory is a normative model, not
a prescriptive one. If we tried to calculate expected utilities for every decision we
make, we would spend our whole lives making calculations. Instead of doing this,
we adopt prescriptive rules of various sorts, including rules of personal behavior and
rules of morality. If these rules are good ones, they will usually prescribe the same
decisions that we would make if we had time to carry out a more thorough analysis.
Other examples of comparison of errors
The basic idea of Pascal’s wager was that some errors are much worse than others.
Analogous situations occur in the courtroom, as illustrated by the quotation that
begins this chapter. We find it worse to convict innocent people of crimes than to
let the guilty go free. As a result, we require a high standard of proof in order to
convict someone. In common law, it is usually said that a person must be guilty “beyond
a reasonable doubt.” Expected utility theory can help us capture this idea more
precisely. Consider the following table:
State of the world
Option Innocent Guilty
Convict −100 0
Acquit 0 −10
The numbers in the cells represent utilities. We assume that convicting the guilty
and acquitting the innocent have utilities of 0. We could, of course, add a constant
to each column, and no conclusion would change. The table indicates that a false
conviction is ten times worse than a false acquittal. What does this ratio imply about
the appropriate interpretation of “beyond a reasonable doubt?” In particular, at what