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or bad according as the outcome is "favourable" or "unfavourable".
In a decision making problem one therefore has to compare the
values of different outcomes. In some cases this comparison can
be based directly on e.g. sums of money to be lost or gained. But
the same sum of money does not have the same value to every
person, moreover, preferences are influenced by taste, ethics etc.
Therefore the concept of utility is introduced, which is a quality
of an outcome that represents its value to a person, i.e. what it is
worth to him. The modern notion of utility was introduced by
J. von Neumann and O. Morgenstern [7], essentially to provide
a way to treat preferences numerically. It is intuitively clear what
it means if a person states that he prefers outcome A to outcome
B and outcome B to outcome C. But without a further theoretical
framework it is impossible to discuss e.g. differences of preference
in a consistent way: the intuitive notion of preference admits only
of a primitive method of measurement, viz. an ordering. The
construction of such a framework, the axiomatically structured
theory of utility, is given in [7], with a penetrating discussion of
the background. It can be seen as a mathematical model describing
properties of a person's preferences. We shall try to sketch the
theory briefly. In doing so we shall follow [3] in calling the possible
outcome of an action a prospect. We shall denote prospects by 0,
possibly with an index. A crucial point in the theory is formed by
the introduction of combined prospects which form a "weighted
average" of other prospects. Let p be a probability; we can consider
the prospect 0 whereby the individual faces the prospect Oi with
probability p and the prospect O2 with probability i-p. Then 0 is
called a mixture of Oi and O2. The following assumptions are
made [3]
1. A person facing two prospects Ox and 02 is able to decide
whether he prefers Ox to 02, whether he prefers O2 to Ox, or
whether he likes them equally well.
2. If he regards Ox at least as well as 02 and 02 at least as well
as O3, then he regards 01 at least as well as 03.
3. If Ox is preferred to O2, which is preferred to 03, then there
is a mixture of Ox and 03 which is preferred to O2 and there is
a mixture of Ox and 03 over which 02 is preferred.
4. Let the person prefer Ox to O2 and let 03 be another prospect.
Then he will prefer a mixture of Ox and 03 (with probabilities
p and 1 -p) to the mixture of 02 and O3 with the same proba
bilities p and 1 -p.
If it can be shown that if an individual's tastes satisfy these
assumptions, he has a utility function u on the set of prospects to
the set of numbers. If the individual faces prospects Ox, O2 etc.,
then to each prospect corresponds a number w(Oj), which is
