If the preferences have been determined in a way consistent with the
axioms, then it can be shown that a *utility function* always
exists. This means that there exists a function
such that, for all
,

if and only if | (9.87) |

in which denotes the expected value of , which is being treated as a random variable under the probability distribution . The existence of implies that it is safe to determine the best action by maximizing the expected utility.

A reward function can be defined using a utility function, , as . The utility function can be converted to a cost function as . Minimizing the expected cost, as was recommended under Formulations 9.3 and 9.4 with probabilistic uncertainty, now seems justified under the assumption that was constructed correctly to preserve preferences.

Unfortunately, establishing the existence of a utility function does not produce a systematic way to construct it. In most circumstances, one is forced to design by a trial-and-error process that involves repeatedly checking the preferences. In the vast majority of applications, people create utility and cost functions without regard to the implications discussed in this section. Thus, undesirable conclusions may be reached in practice. Therefore, it is important not to be too confident about the quality of an optimal decision rule.

Note that if worst-case analysis had been used, then most of the problems discussed here could have been avoided. Worst-case analysis, however, has its weaknesses, which will be discussed in Section 9.5.3.

Steven M LaValle 2020-08-14