To meet the goal of designing a utility function, it turns out that
the preferences must follow rules called the *axioms of
rationality*. They are sensible statements of consistency among the
preferences. As long as these are followed, then a utility function
is guaranteed to exist (detailed arguments appear in
[268,831]). The decision maker is considered *rational* if the following axioms are
followed when defining and :^{9.6}

- If
, then either
or
.

``You must be able to make up your mind.'' - If
and
, then
.

``Preferences must be transitive.'' - If
, then
(9.84)

for any and .

``Evenly blending in a new distribution does not alter preference.'' - If
, then there exists some
and
such that

and

``There is no heaven or hell.''

Each axiom has an intuitive interpretation that makes practical sense.
The first one simply indicates that the preference direction can
always be inferred for a pair of distributions. The second axiom
indicates that preferences must be transitive.^{9.7} The last two axioms are somewhat more
complicated. In the third axiom, is strictly preferred to
. An attempt is made to cause confusion by blending in a third
distribution, . If the same ``amount'' of is blended into
both and , then the preference should not be affected. The
final axiom involves , , and , each of which is
strictly better than its predecessor. The first equation,
(9.85), indicates that if is strictly better than
, then a tiny amount of can be blended into , with
remaining preferable. If had been like ``heaven'' (i.e.,
infinite reward), then this would not be possible. Similarly,
(9.86) indicates that a tiny amount of can be blended
into , and the result remains better than . This means that
cannot be ``hell,'' which would have infinite negative
reward.^{9.8}

Steven M LaValle 2020-08-14