The theory of expected Utility by J. Von Neumann and O. Morgenstern

6.Probability and numerical utilities
Following this idea, only little more effort than using the assumptions of
indifference curve analysis is needed to achieve a numerical utility. First and
foremost, the numerical utility requires the possibility to compare the differences
in utilities. It is a bigger assumption than sole ability to state preferences. Forfurther analysis, a few assumptions will be made. Firstly, let us assume that an
individual has a surjective and complete system of preferences i.e. for any two
objects (or imagined events) he has a unequivocally defined intuition of
preference. In other words, when faced with two alternative events (possibilities)
he is able to clearly state which one of these two he prefers. A natural extension
of this assumption would be the possibility for the individual to compare not only
single events but also combinations of events with attached probabilities. Such
extension is needed for application to economy since many economic activities
are explicitly dependent on probability - which is usually unknown or hard to
estimate (the simplest example - insurance).
Let us assume the following situation. Let three events be denoted by A, B
and C. For the sake of simplicity let the probability of occurrence of events B and
C be equal to 50% i.e. the probability of B occurring is equal 50% and if B does not
occur, than event C must occur with the remaining probability (which in this case
is 50%). Two further assumptions are made considering this situation. Firstly,
the two alternatives B and C are mutually exclusive so there is no possibility of
complementarity. Secondly, we assume absolute certainty of the occurrence of
either event B or C.
In our example we expect the individual to have a clear intuition whether
he prefers event A to the 50 - 50 combination of events B and C or the opposite
(the combination of B and C to the event A). Having established the example,
let us consider three cases. When the individual prefers event A to event B and
at the same time event A to event C (using modern game theory nomenclature:
A ­ B ^ A ­ C), it is clear that he will also prefer event A to the combination of
events B and C. Similarly, if he prefers event B to event A and at the same time
event C to event A (B ­ A ^ C ­ A), he will prefer the combination of events B
and C to the event A. However, if he should prefer event A to let us say B but
at the same time C to A (A ­ B ^ C ­ A), than any statement of his preference
of A to the combination of B and C in such case gives us a fundamentally new
information. Hence, this case provides a base for numerical estimation of the fact
that his preference of A over B is "greater" than his preference of C over A.
The above case can be explained by the use of a very simple example.
Let us assume that an individual prefers a glass of tea to a cup of coffee and at
the same time that he prefers a cup of coffee to a glass of milk. In order to get
to know if the second preference (i.e. difference in utilities) is greater than the
first one, it is enough to make him decide whether he prefers a cup of coffee to a
glass which content will be determined by a toss of a fair coin (heads = tea, tails
= milk).