called the utility of the prospect Oj. The properties of this function
are:
1. u{01) u{0z) if and only if the individual prefers 01 to 02.
2. If 0 is the prospect where, with probability p the individual
faces Oi and with probability i-ft he faces 0%, then
u(0) ft u(0i) x-ftu(0f).
Much has been written about the concept of utility, see e.g. the
historical and critical comments reviewed by L. J. Savage in [9].
A very clear exposition is also given by R. Duncan Luce and
H. Raiffa in [5]. It seems very difficult indeed to assign numbers
to prospects, measuring their desirability and taking account of
financial losses or gains, personal satisfaction, ethical values etc.
connected with each prospect. The originators say (of models in
general)"Similarity to reality is needed to make the operations
significant". We shall later return to the difficulties connected with
this requirement in the context of geodesy.
3. The Elements of a Decision ftrohlem
We shall introduce the different elements of a decision making
problem by means of a simple example. Suppose a surveyor is
measuring some distance by tape. He makes a direct and a reverse
measurementthere will be a difference between the two measure
ments, which we will denote by t. From experience he knows that
under the prevailing circumstances the measuring procedure
results in a standard deviation of 2 f2 cm in a single measurement,
and that the two measurements are stochastically independent.
Now there may be a gross error in one of the measurements. For
reasons of exposition we make the very much simplifying assump
tion that he has either made a gross error of 10 cm in one of the
measurements, or no gross error. These are the two possible states
of nature. The only indication of the possible presence of a gross
error is the size of t, which is a realization of the observation variate
t. The determination of t (or in fact the second measurement) can
be seen as an experiment to get information on the state of nature.
There are two actions he can take, after having found t: remeasure
or not remeasure the distance. Each of these actions will have
certain consequences: remeasuring takes time, but enables him to
remedy a gross error if he has made one. If he has not made a gross
error, his time will be lost. On the other hand, if he does not check
the distance once again, his result may be false. These consequences
are reflected in the losses of utility associated with each action
under each state of nature.
Our surveyor will let it depend on the size of t whether he will
remeasure or not. For certain values of t he will decide to remeasure,
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